Quantum resource estimation using reparameterization methods

Abstract

A system, computer-implemented method, and computer program product are provided that facilitate estimation of quantum resources for calculating an expectation value of a stochastic process using a reparameterization method. According to one aspect, the system may include a processor that executes computer-executable components stored in a memory. The computer-executable components may include a reparameterization component that applies a quantum fault-tolerant operation to a variationally prepared quantum state corresponding to a probability distribution to generate a quantum state corresponding to a target probability distribution. The computer-executable components may further include an estimation component that estimates at least one defined criterion of a quantum computer that will be used to calculate the expectation value of the stochastic process associated with the target probability distribution.

Description

[Technical Field]

[0001] The subject disclosure relates to estimating quantum resources for calculating the expectation value of a stochastic process, and more particularly, to estimating quantum resources for calculating the expectation value of a stochastic process using reparameterization methods. Summary of the Invention

[0002] The following presents a summary to provide a basic understanding of one or more aspects of the invention. This summary is not intended to identify key or critical elements or to delineate the scope of particular aspects or the claims. Its sole purpose is to present concepts in a simplified form as a prelude to the more detailed description that is presented later. In one or more aspects described herein, systems, devices, computer-implemented methods, and / or computer program products are described that can facilitate estimation of quantum resources for calculating the expectation value of a stochastic process using a re-parameterization method.

[0003] According to an aspect, a system can include a processor that executes computer-executable components stored in a memory. The computer-executable components can include a reparameterization component that applies a quantum fault-tolerant operation to a variationally prepared quantum state corresponding to a probability distribution to generate a quantum state corresponding to a target probability distribution. The computer-executable components can further include an estimation component that estimates at least one defined criterion for a quantum computer used to calculate an expectation value of a stochastic process associated with the target probability distribution.

[0004] According to another aspect, the computer-implemented method can include applying, by a system operatively coupled to a processor, a quantum fault-tolerant operation to a variationally prepared quantum state corresponding to a probability distribution to generate a quantum state corresponding to a target probability distribution. The computer-implemented method can further include estimating, by the system, at least one defined criterion for a quantum computer used to calculate an expectation value of a stochastic process associated with the target probability distribution.

[0005] According to another aspect, a computer program product includes a computer-readable storage medium having program instructions embodied thereon, the program instructions executable by a processor to cause the processor to apply a quantum fault-tolerant operation to a variationally prepared quantum state corresponding to a probability distribution to generate a quantum state corresponding to a target probability distribution, the program instructions further executable by the processor to cause the processor to estimate at least one defined criterion for a quantum computer used to calculate an expectation value of a stochastic process associated with the target probability distribution. [Brief explanation of the drawings]

[0006] [Figure 1] FIG. 1 illustrates a block diagram of an example, non-limiting system that can facilitate estimation of quantum resources for calculating the expectation value of a stochastic process using a reparameterization method according to one or more aspects described herein. [Figure 2] FIG. 1 illustrates a block diagram of an example, non-limiting system that can facilitate estimation of quantum resources for calculating the expectation value of a stochastic process using a reparameterization method according to one or more aspects described herein. [Figure 3]FIG. 1 illustrates a block diagram of an example, non-limiting system that can facilitate estimation of quantum resources for calculating the expectation value of a stochastic process using a reparameterization method according to one or more aspects described herein. [Figure 4] FIG. 1 illustrates an example, non-limiting graph that can facilitate estimation of quantum resources for calculating the expectation value of a stochastic process using a reparameterization method in accordance with one or more aspects described herein. [Figure 5] FIG. 1 illustrates an example, non-limiting circuit that can facilitate estimation of quantum resources for calculating the expectation value of a stochastic process using a reparameterization method in accordance with one or more aspects described herein. [Figure 6A] FIG. 1 illustrates an example, non-limiting graph that can facilitate estimation of quantum resources for calculating the expectation value of a stochastic process using a reparameterization method in accordance with one or more aspects described herein. [Figure 6B] FIG. 1 illustrates an example, non-limiting graph that can facilitate estimation of quantum resources for calculating the expectation value of a stochastic process using a reparameterization method in accordance with one or more aspects described herein. [Figure 6C] FIG. 1 illustrates an example, non-limiting graph that can facilitate estimation of quantum resources for calculating the expectation value of a stochastic process using a reparameterization method in accordance with one or more aspects described herein. [Figure 7A] FIG. 1 illustrates an example, non-limiting graph that can facilitate estimation of quantum resources for calculating the expectation value of a stochastic process using a reparameterization method in accordance with one or more aspects described herein. [Figure 7B] FIG. 1 illustrates an example, non-limiting graph that can facilitate estimation of quantum resources for calculating the expectation value of a stochastic process using a reparameterization method in accordance with one or more aspects described herein. [Figure 8A] FIG. 1 illustrates an example, non-limiting graph that can facilitate estimation of quantum resources for calculating the expectation value of a stochastic process using a reparameterization method in accordance with one or more aspects described herein. [Figure 8B] FIG. 1 illustrates an example, non-limiting graph that can facilitate estimation of quantum resources for calculating the expectation value of a stochastic process using a reparameterization method in accordance with one or more aspects described herein. [Figure 9A] FIG. 1 illustrates an example, non-limiting graph that can facilitate estimation of quantum resources for calculating the expectation value of a stochastic process using a reparameterization method in accordance with one or more aspects described herein. [Figure 9B] FIG. 1 illustrates an example, non-limiting graph that can facilitate estimation of quantum resources for calculating the expectation value of a stochastic process using a reparameterization method in accordance with one or more aspects described herein. [Figure 9C] FIG. 1 illustrates an example, non-limiting graph that can facilitate estimation of quantum resources for calculating the expectation value of a stochastic process using a reparameterization method in accordance with one or more aspects described herein. [Figure 10] FIG. 1 illustrates a flow diagram of an example, non-limiting, computer-implemented method that can facilitate estimation of quantum resources for calculating the expectation value of a stochastic process using a reparameterization method according to one or more aspects described herein. [Figure 11] FIG. 1 illustrates a block diagram of an exemplary non-limiting operating environment that can facilitate one or more aspects described herein. [Figure 12] FIG. 1 illustrates a block diagram of an exemplary non-limiting cloud computing environment in accordance with one or more aspects of the subject disclosure. [Figure 13]FIG. 1 illustrates a block diagram of example non-limiting abstraction model layers in accordance with one or more aspects of the subject disclosure. DETAILED DESCRIPTION OF THE INVENTION

[0007] The following detailed description is merely exemplary and is not intended to limit the embodiments and / or the application or uses of the embodiments, and further, there is no intention to be bound by any express or implied information presented in the preceding Background or Summary section or in the Detailed Description section.

[0008] One or more aspects will now be described with reference to the drawings, wherein like reference numerals are used to refer to like elements throughout. In the following description, for purposes of explanation, numerous specific details are set forth in order to provide a more thorough understanding of one or more aspects. It will be apparent, however, that in various instances, one or more aspects may be practiced without these specific details.

[0009] As referred to herein, a “derivative” and / or “derivative asset” is a contract between an issuer and a holder that is valid until its expiration date. As referred to herein, an “entity” may include a human, a client, a user, a computing device, a software application, an agent, a machine learning (ML) model, an artificial intelligence (AI), and / or another entity. As used herein, when an element is referred to as being “coupled” to another element, it will be understood that one or more different types of bonds can be described, including, but not limited to, a chemical bond, a communicative bond, an electrical bond, an electromagnetic bond, a operative bond, an optical bond, a physical bond, a thermal bond, and / or another type of bond.

[0010] 1, 2, and 3 illustrate block diagrams of exemplary, non-limiting systems 100, 200, and 300, respectively, that can facilitate quantum resource estimation for calculating the expectation value of a stochastic process using a reparameterization method according to one or more embodiments described herein. Systems 100, 200, and 300 can each comprise a quantum resource estimation system 102. The quantum resource estimation system 102 of system 100 illustrated in FIG. 1 can comprise a memory 104, a processor 106, a reparameterization component 108, an estimation component 110, and / or a bus 112. The quantum resource estimation system 102 of system 200 illustrated in FIG. 2 can further comprise a variational component 202. The quantum resource estimation system 102 of system 300 depicted in FIG. 3 can further comprise an error analysis component 302.

[0011] While some aspects of the subject disclosure describe exemplary applications of quantum resource estimation system 102 for estimating quantum resources to calculate the expected value of a stochastic process, such as a derivative asset, it should be understood that the subject disclosure is not so limited. For example, quantum resource estimation system 102 may estimate quantum resources to calculate the expected value of another stochastic process (e.g., any type of stochastic process).

[0012] It should be understood that aspects of the subject disclosure illustrated in the various figures disclosed herein are for illustrative purposes only, and thus the architecture of such aspects is not limited to the systems, devices, and / or components shown therein. For example, in some aspects, system 100, system 200, system 300, and / or quantum resource estimation system 102 may further comprise various computers and / or computing-based elements described herein with reference to operating environment 1100 and FIG. 11. In some aspects, such computers and / or computing-based elements may be used in connection with implementing one or more of the systems, devices, components, and / or computer-implemented operations shown and described in connection with FIG. 1, FIG. 2, FIG. 3, and / or other figures disclosed herein.

[0013] Memory 104 may store one or more computer- and / or machine-readable, writable, and / or executable components and / or instructions that, when executed by processor 106 (e.g., a classical processor, a quantum processor, and / or another type of processor), can facilitate performance of operations defined by the executable components and / or instructions. For example, memory 104 may store computer- and / or machine-readable, writable, and / or executable components and / or instructions that, when executed by processor 106, can facilitate performance of various functions described herein related to quantum resource estimation system 102, reparameterization component 108, estimation component 110, variational component 202, error analysis component 302, and / or another component associated with quantum resource estimation system 102 described herein with or without reference to various figures of the subject disclosure.

[0014] The memory 104 may include volatile memory (e.g., random access memory (RAM), static RAM (SRAM), dynamic RAM (DRAM), and / or another type of volatile memory) and / or non-volatile memory (e.g., read-only memory (ROM), programmable ROM (PROM), electrically programmable ROM (EPROM), electrically erasable programmable ROM (EEPROM), and / or another type of non-volatile memory), which may utilize one or more memory architectures. Further examples of memory 104 are described below with reference to system memory 1116 and FIG. 11 . Such examples of memory 104 may be utilized to implement any aspect of the subject disclosure.

[0015] Processor 106 may include one or more types of processors and / or electronic circuitry (e.g., classical processors, quantum processors, and / or other types of processors and / or electronic circuitry) that can implement one or more computer- and / or machine-readable, writable, and / or executable components and / or instructions that may be stored in memory 104. For example, processor 106 may perform various operations that may be specified by such computer- and / or machine-readable, writable, and / or executable components and / or instructions, including, but not limited to, logic, control, input / output (I / O), and / or arithmetic. In some aspects, processor 106 may include one or more central processing units, multi-core processors, microprocessors, dual microprocessors, microcontrollers, systems-on-chips (SOCs), array processors, vector processors, quantum processors, and / or other types of processors. Further examples of processor 106 are described below with reference to processing unit 1114 and FIG. 11 . Such an example of the processor 106 may be utilized to implement any aspect of the subject disclosure.

[0016] The quantum resource estimation system 102, memory 104, processor 106, reparameterization component 108, estimation component 110, variational component 202, error analysis component 302, and / or other components of the quantum resource estimation system 102 as described herein may be communicatively, electrically, operatively, and / or optically coupled to each other via a bus 112 to perform the functions of the system 100, the system 200, the system 300, the quantum resource estimation system 102, and / or any components coupled thereto. The bus 112 may include one or more memory buses, memory controllers, peripheral buses, external buses, local buses, quantum buses, and / or other types of buses that may utilize various bus architectures. Further examples of the bus 112 are described below with reference to the system bus 1118 and FIG. 11 . Such examples of the bus 112 may be utilized to implement any aspect of the subject disclosure.

[0017] The quantum resource estimation system 102 may include any type of component, machine, device, facility, apparatus, and / or equipment that includes a processor and / or is capable of effective and / or operable communication with wired and / or wireless networks. All such aspects are contemplated. For example, the quantum resource estimation system 102 may include a server device, a computing device, a general-purpose computer, a special-purpose computer, a quantum computing device (e.g., a quantum computer), a tablet computing device, a handheld device, a server-class computing machine and / or database, a laptop computer, a notebook computer, a desktop computer, a mobile phone, a smartphone, a consumer electronics product and / or appliance, an industrial and / or commercial device, a digital assistant, a multimedia Internet-enabled telephone, a multimedia player, and / or another type of device.

[0018] The quantum resource estimation system 102 may be coupled (e.g., communicatively, electrically, operatively, optically, and / or via another type of coupling) to one or more external systems, sources, and / or devices (e.g., classical and / or quantum computing devices, communication devices, and / or another type of external system, source, and / or device) using wires and / or cables. For example, the quantum resource estimation system 102 may be coupled (e.g., communicatively, electrically, operatively, optically, and / or via another type of coupling) to one or more external systems, sources, and / or devices (e.g., classical and / or quantum computing devices, communication devices, and / or another type of external system, source, and / or device) using data cables, including, but not limited to, high-definition multimedia interface (HDMI) cables, recommended standard (RS) 232 cables, Ethernet cables, and / or other data cables.

[0019] In some aspects, quantum resource estimation system 102 may be coupled (e.g., communicatively, electrically, operatively, optically, and / or via another type of coupling) to one or more external systems, sources, and / or devices (e.g., classical and / or quantum computing devices, communication devices, and / or another type of external system, source, and / or device) via a network. For example, such a network may include wired and / or wireless networks, including, but not limited to, a cellular network, a wide area network (WAN) (e.g., the Internet), a local area network (LAN), and / or another network. The quantum resource estimation system 102 may communicate with one or more external systems, sources, and / or devices, e.g., computing devices, using substantially any desired wired and / or wireless technology, including, but not limited to, Wireless Fidelity (Wi-Fi), Global System for Mobile Communications (GSM), Universal Mobile Telecommunications System (UMTS), Worldwide Interoperability for Microwave Access (WiMAX), Enhanced General Packet Radio Service (Enhanced GPRS), Third Generation Partnership Project (3GPP), Long Term Evolution (LTE), and other technologies. ), 3rd Generation Partnership Project 2 (3GPP2), Ultra Mobile Broadband (UMB), High Speed ​​Packet Access (HSPA), Zigbee and other 802.XX wireless technologies and / or legacy communication technologies, BLUETOOTH®, Session Initiation Protocol (SIP), ZIGBEE®, RF4CE protocol, WirelessHART protocol, 6LoWPAN (IPv6 over Low Power Wireless Area Network), Z-Wave, ANT, Ultra Wideband (UWB) standard protocols, and / or other proprietary and non-proprietary communication protocols.Thus, in some aspects, the quantum resource estimation system 102 may include hardware (e.g., a central processing unit (CPU), a transceiver, a decoder, quantum hardware, a quantum processor, and / or other hardware), software (e.g., a set of threads, a set of processes, running software, a quantum pulse schedule, a quantum circuit, a quantum gate, and / or other software), or a combination of hardware and software that can facilitate communication of information between the quantum resource estimation system 102 and external systems, sources, and / or devices (e.g., a computing device, a communication device, and / or another type of external system, source, and / or device).

[0020] Quantum resource estimation system 102 may include one or more computer- and / or machine-readable, writable, and / or executable components and / or instructions that, when executed by processor 106 (e.g., a classical processor, a quantum processor, and / or another type of processor), can facilitate performance of operations defined by such components and / or instructions. Furthermore, in many aspects, any component associated with quantum resource estimation system 102 as described herein with or without reference to various figures of the subject disclosure can include one or more computer- and / or machine-readable, writable, and / or executable components and / or instructions that, when executed by processor 106, can facilitate performance of operations defined by such components and / or instructions. For example, the reparameterization component 108, the estimation component 110, the variational component 202, the error analysis component 302, and / or any other components associated with (e.g., communicatively, electronically, operatively, and / or optically coupled to, and / or utilized by) the quantum resource estimation system 102 as disclosed herein may include such computer- and / or machine-readable, writable, and / or executable components and / or instructions.As a result, according to many aspects, the quantum resource estimation system 102 and / or any components associated therewith as disclosed herein may utilize the processor 106 to execute such computer- and / or machine-readable, writable, and / or executable components and / or instructions to facilitate performance of one or more operations described herein with reference to the quantum resource estimation system 102 and / or any such components associated therewith.

[0021] The quantum resource estimation system 102 may facilitate (e.g., via the processor 106) the execution of operations performed by and / or associated with the reparameterization component 108, the estimation component 110, the variational component 202, the error analysis component 302, and / or another component associated with the quantum resource estimation system 102 as disclosed herein. For example, as described in detail below, the quantum resource estimation system 102 may facilitate (e.g., via the processor 106) applying a quantum fault-tolerant operation to a variationally prepared quantum state corresponding to a probability distribution to generate a quantum state corresponding to a target probability distribution and / or estimating at least one defined criterion for a quantum computer used to calculate an expectation value of a stochastic process associated with the target probability distribution.

[0022] In another example, as described in detail below, the quantum resource estimation system 102 may further facilitate (e.g., via the processor 106) applying a transformation operation to the variationally prepared quantum state to generate a quantum state corresponding to a target probability distribution having a defined mean of the target probability distribution, a defined standard deviation of the target probability distribution, and / or one or more explicit parameters specifying the target probability distribution; applying a quantum fault-tolerant operation to the variationally prepared quantum state to prepare the quantum state as a superposition for possible paths of a discrete-time multivariate stochastic process; training a variational quantum circuit to prepare the variationally prepared quantum state and to reduce the computational cost of quantum arithmetic operations performed by the quantum computer to calculate the expectation of the stochastic process associated with the target probability distribution; training the variational quantum circuit using a Hamiltonian operator to generate a basis state corresponding to the target probability distribution; and calculating one or more errors associated with at least one of applying the quantum fault-tolerant operation to the variationally prepared quantum state to generate a quantum state, estimating at least one defined criterion, or calculating the expectation of the stochastic process associated with the target probability distribution.

[0023] In the above example, the at least one defined criterion may include attributes, conditions, properties, parameters, and / or configurations of the quantum computer that enable the quantum computer to achieve the defined quantum supremacy in calculating the expectation value of the stochastic process associated with the target probability distribution. In the above example, the probability distribution may include a standard normal probability distribution, and the target probability distribution may include a normal probability distribution.

[0024] According to various aspects of the subject disclosure described herein, to perform one or more of the operations described above, quantum resource estimation system 102, reparameterization component 108, estimation component 110, variational component 202, and / or error analysis component 302 may define and / or implement one or more of the algorithms (e.g., Algorithms 2.1, 3.1, 3.2, 4.1, and / or 4.2) and / or one or more of the equations (e.g., Equations (1)-(109)) described below with reference to Sections 1.0-11.0. For example, with reference to Sections 3.2 and 10.0 described below, reparameterization component 108 may apply a quantum fault-tolerant operation to a variationally prepared quantum state corresponding to a probability distribution (e.g., a standard normal probability distribution) to generate a quantum state corresponding to a target probability distribution (e.g., a normal probability distribution). In this example, the reparameterization component 108 can apply a quantum fault-tolerant operation to the variationally prepared quantum state to prepare the quantum state as a superposition for possible paths of a discrete-time multivariate stochastic process. In this example, the reparameterization component 108 can apply a transformation operation to the variationally prepared quantum state to generate a quantum state corresponding to a target probability distribution having at least one of a defined mean of the target probability distribution, a defined standard deviation of the target probability distribution, or one or more explicit parameters that specify the target probability distribution.

[0025] In another example, and referring to sections 3.1.2 and 3.2.3 described below, estimation component 110 can estimate at least one defined criterion for a quantum computer used to calculate the expected value of a stochastic process associated with a target probability distribution. For example, estimation component 110 can estimate at least one defined criterion for a quantum computer used to calculate the value of an asset, such as a derivative asset, associated with the target probability distribution. For example, estimation component 110 can estimate at least one defined criterion, including, but not limited to, attributes, conditions, properties, parameters, configurations, and / or another criterion of the quantum computer, that enable the quantum computer to achieve a defined quantum supremacy in calculating the expected value of a stochastic process associated with the target probability distribution (e.g., the value of a derivative asset).

[0026] In another example, see Sections 3.2.1 and 11.0, described below, variational component 202 can train a variational quantum circuit to prepare a variationally prepared quantum state. For example, variational component 202 can train a variational quantum circuit to prepare a variationally prepared quantum state by training the variational quantum circuit using a Hamiltonian operator to generate a basis state corresponding to a target probability distribution. It should be understood that variational component 202 can train a variational quantum circuit to prepare a variationally prepared quantum state to reduce the computational cost of quantum arithmetic operations performed by a quantum computer to calculate the expected value of a stochastic process associated with the target probability distribution (e.g., the value of a derivative asset).

[0027] In another example, and referring to Section 3.2.2 below, the error analysis component 302 may calculate one or more errors associated with at least one of applying a quantum fault-tolerant operation to a variationally prepared quantum state to generate a quantum state, estimating at least one defined criterion, or calculating an expected value of a stochastic process associated with a target probability distribution (e.g., the value of a derivative asset).

[0028] According to various aspects, as described in the following sections, the quantum resource estimation system 102 can determine an upper bound on the resources (e.g., quantum computing resources) involved in providing a useful quantum advantage in pricing derivatives. To that end, the quantum resource estimation system 102 can provide an initial complete resource estimate for pricing useful quantum derivatives using auto-callable options and target accrual redemption forwards (TARF) derivatives as an exemplary benchmark use case. The quantum resource estimation system 102 can provide a new method (reparameterization method) for pricing quantum derivatives that overcomes and avoids obstacles in known approaches. It should be appreciated that the reparameterization method, which can be defined and / or implemented by the quantum resource estimation system 102 as described in the following sections, hybridizes a pre-trained variational circuit with fault-tolerant quantum computing, significantly reducing the resources (e.g., quantum computing resources) involved in estimating the value of a derivative. As described below, according to various aspects of the subject disclosure, the quantum resource estimation system 102 may determine that autocallable options and TARF derivatives as exemplary benchmark use cases require, for example, approximately 8,000 (8k) logical qubits, an estimated T-depth of approximately 50 million, and an estimated logical clock speed of approximately 10 megahertz (MHz) to achieve a defined quantum supremacy.

[0029] 1.0 Derivatives Pricing

[0030] Pricing derivatives contracts using Monte Carlo methods is computationally intensive in the financial sector, and quantum supremacy in this application is highly valuable. In various aspects, as described in the following sections, the quantum resource estimation system 102 can provide the first detailed resource estimates of the conditions involved in achieving quantum supremacy in derivatives pricing. To accomplish this, the quantum resource estimation system 102 can define and / or implement novel methods, described below, for loading stochastic processes onto a quantum computer in accordance with one or more aspects of the subject disclosure.

[0031] As defined above and referred to herein, a "derivative" and / or "derivative asset" is a contract between an issuer and a holder that is valid until its expiration date. Examples of these derivative assets include, but are not limited to, forward contracts, options, autocallable options, target accrual redemption notes (TARNs), TARFs, and / or another derivative asset. Each derivative has a payoff that defines what the holder will receive. The payoff depends on the value of one or more underlying assets over the contract's life. Examples of these underlying assets include, but are not limited to, stocks, currencies, commodities, and / or another underlying asset. Derivative contracts are widespread in finance, with uses ranging from risk hedging to speculation. The objective of derivative pricing is to determine the current value of a derivative contract given uncertainty about the future value of the underlying asset and, therefore, the payoff.

[0032] The underlying asset is typically modeled as a stochastic process under assumptions such as no arbitrage, which means that a particular asset is not priced differently in different markets, and therefore it is not possible to buy an asset from one market and immediately sell it in another market for a profit. A common model is that the underlying asset evolves under geometric Brownian motion.

[0033]

number

[0034] Let be a vector of values ​​of d underlying assets at time t.

[0035]

number

[0036] Let be the path of the discrete-time multivariate stochastic process that describes the value of those assets. Both notations are used for the path in this document. The corresponding probability density function is

[0037]

number

[0038] It is shown as follows.

[0039]

number

[0040] Let be the payoffs of derivatives on these assets. Equation (1), defined below, can be used to price the derivatives.

[0041]

number

[0042] If the underlying stochastic processes are modeled with geometric Brownian motion, they have the following transition probabilities:

[0043]

number

[0044] where:

[0045]

number

[0046] Equation (2) defined above at time t is

[0047]

number

[0048] In

[0049]

number

[0050] Note that the parameter r and σ depend on the asset vector at time t-1 via j are the risk-free rate and volatility of the jth asset, respectively, Δt is the duration between steps of the stochastic process, and Σ is the d × d positive definite covariance matrix of the d underlying assets.

[0051]

number

[0052] where -1 ≤ ρ ij ≦1 is the correlation between assets i and j.

[0053]

number

[0054] The probability of

[0055]

number

[0056] is.

[0057] The risk-free rate referred to above is the rate of return on an investment in a risk-free asset. While such assets are purely theoretical, Treasury bonds are commonly used to represent such assets, and we approximate r as the yield on a Treasury bond minus the current inflation rate.

[0058] Classically, some simple derivatives in this model are easy to price, for example, European call options, which can be analytically priced using the Black-Scholes equation. Easy-to-price derivatives are often path-independent, and their payoffs are exercise time.

[0059]

number

[0060] This contrasts with path-dependent derivatives, which are more difficult to price and are often priced in practice using classical Monte Carlo methods.

[0061] Using classical Monte Carlo methods, the accuracy of derivatives pricing is

[0062]

number

[0063] where M is the number of samples. Quantum algorithms based on amplitude estimation can generally be used to improve this to O(1 / M). Recent work has explored how to specialize this advantage for option pricing and risk analysis. This is only a quadratic speedup, and the subject disclosure focuses on derivatives complex enough to have a large M in practice. According to various aspects, as described in the following sections, the quantum resource estimation system 102 can provide end-to-end quantum resource estimation for autocallable options and TARFs, two examples of such derivatives, both of which are computationally expensive path-dependent derivatives. In doing so, the quantum resource estimation system 102 (e.g., via the reparameterization component 108, the estimation component 110, and / or the variational component 202) can refine and optimize the loading of the quantum state of the underlying distribution over the asset path. This loading step remains open (e.g., unresolved) in the prior art, and quantum resource estimation system 102 (e.g., via estimation component 110) can provide an initial description of the resources required to accomplish such a loading step.

[0064] In addition to estimating the resources that can be used for path loading, the quantum resource estimation system 102 can also provide several optimizations, including a deliberate shift in calculations from price space to return space and new reparameterization methods that result in significant resource reductions and are summarized in Table 1.

[0065] [Table 1]

[0066] Table 1 shown above shows the 2 x 10 -31 illustrates resources estimated by implementing a quantum resource estimation system 102 in accordance with one or more aspects of the subject disclosure to price derivatives using different methods for a target error of T. In such a quantum resource estimation system 102 implementation, there is a basket of knock-in options and auto-callable (auto) options with five auto-call days, and a TARF with one underlying asset and 26 simulation days. In such a quantum resource estimation system 102 implementation, it was determined that the Grover-Rudolph method is not applicable in practice and that a Riemann summation method can include normalization assumptions to avoid exponentially growing errors in T. As detailed in the Riemann Sum (no-norm) row of Table 1, even when these normalization issues are avoided, a reparameterization method that may be defined and / or implemented by the quantum resource estimation system 102 in accordance with one or more aspects of the subject disclosure performs best. Section 3.1 below describes normalization of Riemann summations, and detailed resource estimates that may be defined and / or implemented by estimation component 110 are described later in sections 3.1.2 and 3.2.3.

[0067] 1.1 Discretized Derivatives Pricing

[0068] To map the derivatives pricing problem to quantum states,

[0069]

number

[0070] is discretized. Classically, this is not very important due to the high precision available, but to study the minimum qubit criteria, discretization is explicitly considered.

[0071] Each value

[0072]

number

[0073] Suppose that ω is discretized into different n-qubit registers, i.e., mapped to a regular lattice. Then, the discrete space of paths can be defined as ω∈Ω. Now, the price expectation is the sum

[0074]

number

[0075] is.

[0076] Here, the probability p(ω) can be defined in several ways. For example, the midpoint of a grid cell is

[0077]

number

[0078] It is possible to obtain that:

[0079] where:

[0080]

number

[0081] is restricted to discrete midpoints. Alternatively, p(ω) can be defined as an integral over discrete cells. These expressions are the same in the fine-grid limit, and in accordance with one or more aspects of the subject disclosure, the midpoint method is used.

[0082] 1.2 Price Space vs. Return Space

[0083] As mentioned above, geometric Brownian motion can be used to model the price of an underlying asset. As referred to herein, this is called a price space description of the underlying stochastic process. In the price space, the transition probabilities are given by a multivariate lognormal distribution.

[0084] An alternative but equivalent representation considers a stochastic process for the log returns of the underlying assets and performs all calculations in the return space. When asset prices follow a log-normal distribution, the log returns are made normally distributed. The vector of underlying log returns for d assets at time t is

[0085]

number

[0086] The transition probabilities can then be given by a multivariate normal distribution.

[0087]

number

[0088] where:

[0089] μ=(μ1,μ2,…μ d ) T , (9)

[0090]

number

[0091] where σ, Δt, Σ, and r are the same Brownian motion parameters as in price space. Note that they are no longer conditioned on their values ​​at previous time steps. In fact, the path distribution in return space contains dT independent Gaussians.

[0092] Note the overloaded notation from the price space formulation, as these representations are interchangeable. At any time t', the price of asset j is

[0093]

number

[0094] can be calculated from the return space using:

[0095] The calculation just described is used when the stochastic process is modeled in return space, but the payoff is defined in terms of asset prices. In various aspects of the subject disclosure described herein, it will be clear from context which space such aspects are operating in.

[0096] Switching between price and return space changes the log-normal loadings to normal loadings. In general, normal loadings are easier because their underlying stochastic evolution is independent of the price at the previous time step, as can be seen by comparing equations (3) and (10) defined above. Therefore, the probability distribution over all T time steps of the stochastic process is

[0097]

number

[0098] teeth,

[0099]

number

[0100] can be simultaneously calculated by

[0101] This advantage can complement the quantum arithmetic used to evaluate the exponent in equation (11) defined above. In various aspects of the subject disclosure, the quantum resource estimation system 102 can take advantage of this advantage by employing the reparameterization methods described herein. Furthermore, when dealing with derivatives where payoffs are directly defined in terms of logarithmic returns and are independent of individual asset prices, this is another reason why the quantum resource estimation system 102 can operate in return space.

[0102] 2.0 Core Approach

[0103] The targeted disclosure approach to derivatives pricing extends quantum average estimation by defining the normalized payoff of any path as

[0104]

number

[0105] Let it be given by:

[0106] Algorithm 2.1, defined below, can proceed in four phases. First, probability distributions are loaded in the form of a superposition over all possible paths. Second, payoffs for all possible paths are computed in quantum parallelism. Third, expected payoffs are stored in the amplitudes of marked states. Fourth, the amplitude estimates are used to retrieve the amplitudes using an O(1 / ε) query for a given target precision ε>0.

[0107] Algorithm 2.1 - Core Approach to Derivatives Pricing

[0108] Use n, d, and T, all of which are positive integers.

[0109] We find an operator P that loads a superposition of probabilistically weighted paths into the register of an ndT qubit.

[0110] 1. Apply operator P to the quantum state

[0111]

number

[0112] Prepare the following.

[0113] 2.

[0114]

number

[0115] quantum register

[0116]

number

[0117] Insert this into the calculation.

[0118] 3. Introducing ancillary qubits,

[0119]

number

[0120] Rotate the value in the register to that amplitude.

[0121]

number

[0122] 4. Use amplitude estimation to extract the probability that the ancillary |1〉, which is the (e.g., discretized) expected payoff

[0123]

number

[0124] This can be rescaled to

[0125]

number

[0126] get.

[0127] Note that steps 1-3 in Algorithm 2.1 load the exact answer after one execution. If it were possible to read the amplitudes directly, the quantum resource estimation system 102 could compute the expected value of all paths with a fixed number of queries. This is unfortunately not possible, so amplitude estimation introduces linear overhead to extract an answer of a given precision. This can be an important conceptual difference from classical Monte Carlo, where samples are taken from paths. In Algorithm 2.1, the quantum resource estimation system 102 can compute (e.g., all) possible paths and take samples (e.g., with estimated amplitudes) of the expected payoff.

[0128] Another distinguishing feature of the quantum approach is that the quantum resource estimation system 102 can normalize the payoff and store it in the amplitude of the state. This normalization can be rescaled at the end, but the error is also scaled up, which can affect the error scaling. In the Riemann summation method described in Section 3.1, this version of normalization rescaling can cause errors to accumulate quickly.

[0129] 2.1 Amplitude Estimation of Derivatives Pricing

[0130] Typically, path-dependent derivatives such as autocallables and TARFs are priced using Monte Carlo methods.

[0131]

number

[0132] is derived by modeling the underlying stochastic process, and the expected payoff is then calculated using the estimator.

[0133]

number

[0134] This estimator has an error of ε=O(M -1 / 2 ) converges to the true expected value.

[0135] This convergence is confirmed by using the quantum amplitude estimation for Monte Carlo: ε = O(M -1 ) can be quadratically accelerated to

[0136]

number

[0137] It takes as input a unitary operator A for n+1 qubits, such that

[0138] Here, the parameter a is unknown, and the last qubit acts as a label to distinguish the |Ψ0〉 state from the |Ψ1〉 state.

[0139] The amplitude estimation is performed using the operator (often called the Grover operator)

[0140]

number

[0141] a is determined by repeated application of

[0142]

number

[0143] and

[0144]

number

[0145] is the reflection operator. By using phase estimation and the quantum Fourier transform, a can be calculated in O(M -1 ) can be determined with an accuracy of . Unfortunately, the depth of the resulting quantum circuit is O(1 / ε), which requires the use of resource-expensive quantum Fourier transforms. Recent developments have introduced other approaches that aim to reduce the resources required to perform amplitude estimation and can eliminate the quantum phase estimation.

[0146] The most efficient variant of amplitude estimation known to date is Iterative Quantum Amplitude Estimation (IQAE). It has been empirically shown that IQAE outperforms other known variants. It omits quantum phase estimation, yet achieves performance four times better than canonical phase estimation approaches. Furthermore, for practical considerations, the following bounds have been shown to hold:

[0147]

number

[0148] where:

[0149]

number

[0150] denotes the worst-case number of oracle invocations, i.e., applications of Q, to achieve an estimation error of ε>0 with confidence 1-α,α∈(0,1).

[0151] 2.2 Route Distribution Loading

[0152] For Algorithm 2.1 to achieve practical quantum supremacy, the path loadings and resources for calculating the payoffs are taken into account. In some cases, analytical forms exist that can simplify the path loadings. For example, for path-independent derivatives, the distribution over the path is not relevant. What is relevant is the distribution of the final underlying price S, such as the lognormal distribution given by the Black-Scholes model. T , which means that the distribution can be analytically calculated and variationally or explicitly loaded into a quantum state. Unfortunately, with quantum dots, the analytical form of this distribution means that these derivatives are typically classically simple. Thus, in accordance with various aspects of the subject disclosure, the quantum resource estimation system 102 may focus on path-dependent derivatives where superpositions over paths are calculated.

[0153] Loading general distributions is exponentially difficult, but several methods have been proposed. Grobel-Rudolph's method exists as an efficient quantum algorithm for loading when the distribution is efficiently integrable. However, this algorithm has limited applicability to derivatives pricing in practice because the associated probability distributions are quantum but still involve Monte Carlo integration, which can simply be avoided using amplitude estimation. The drawbacks of this method are discussed in more detail in Section 7.0.

[0154] Another approach to loading path distributions involves the use of quantum generative adversarial networks (qGANs), which are attractive because they reduce the loading overhead, but it is not yet clear how to predict what the training overhead of a given qGAN will be in practice.

[0155] 2.3 Error analysis

[0156] In this section, we consider the various factors that contribute to the overall error in quantum approaches to option pricing. There are three main factors that contribute to the error in the approach in Algorithm 2.1: δ =f max -f min Let's say.

[0157] Truncation Error: The price of a derivative is determined by the integral over all possible values ​​of the underlying price or return. It is infeasible to calculate the integral over an infinite domain, and therefore the quantum resource estimation system 102 may limit the domain of the integral as follows: the price and / or logarithmic return is determined over the range [B l ,B u ]. This restriction of the domain excludes the probability mass of α. The density function at each step is P max There is an upper limit of f δ If there is an upper limit of

[0158]

number

[0159] A truncation error may occur, which is given by:

[0160] Discretization error: This error (ε disc The error (denoted by ) arises from using a Riemann sum over a finite lattice of points to approximate the integral. This error can be reduced by increasing the number of qubits (n) to approximate the sum.

[0161] Amplitude estimation error: Amplitude estimation is 1 / ε of the state preparation procedure and price calculation. amp When iterations are used, ε amp An error occurs.

[0162] Truncation and discretization errors are discussed in more detail below.

[0163] 2.3.1 Truncation Error

[0164] In this section, we present the truncation error in the return space and then directly extend it to the price space. max Using Chernoff tail bounds on the Gaussian, the logarithmic return of asset i falls within the interval [μ i -wσ max ,μ i +wσ max The probability that it is outside

[0165]

number

[0166] The union bound ensures that any logarithmic return (e.g., d assets over T time steps) falls within the interval

[0167]

number

[0168] The probability that it is outside is

[0169]

number

[0170] The initial asset price is bounded by

[0171]

number

[0172] Then the corresponding interval in price space is

[0173]

number

[0174] is given by

[0175] The quantum resource estimation system 102 can then define a truncated window of dT distinct n-qubit register values ​​that are w standard deviations around the mean value for each time step. The truncation error can then be calculated as:

[0176]

number

[0177] is given by

[0178] 2.3.2 Discretization error

[0179] The final output of the amplitude estimation algorithm represents a Riemann sum that approximates a truncated multidimensional integral. The integral is over dT variables corresponding to d assets at T time steps. Each underlying asset and / or return is measured over the interval [B l ,B u ]. To calculate the discretization error, we apply a multidimensional variation of the midpoint formula as follows: Suppose there are n qubits used to represent each underlying asset, and the domain is 2 ndT cells, and for each value of the register we associate the value of the integrand at the midpoint of the corresponding cell. Assume that β provides an upper bound on the second derivative of the integrand (e.g., this means that the deviation from linearity over a range of length l is βl 2 / 2).

[0180] Consider the error from discretization accumulated over a single cell. Each cell has side lengths (B u -B l ) / 2 nand is a hypercube of dimension l. Note that by symmetry, the linear component of the deviation from the value at the cell center integrates to zero over the cell. Thus, the error at each cell is the length of the edge centered at the origin, l = (B u -B l ) / 2 n The term βx on the dT hypercube 2 It can be bounded by the integral of / 2.

[0181]

number

[0182] Aggregating the errors across all cells is

[0183]

number

[0184] to provide.

[0185] The number of standard deviations used in discretization and the maximum eigenvalue σ of the covariance matrix max With respect to and, the total discretization error is

[0186]

number

[0187] It is bounded by

[0188] For the target discretization error, Eq. (23) further becomes:

[0189]

number

[0190] gives the total number of qubits that can be used to represent d assets for T time steps, given by

[0191] Truncation and discretization errors apply generally to the methods introduced here, but each method has additional method-specific sources of error, which are discussed separately for each approach.

[0192] 3.0 Advantageous Methods for Pricing Quantum Derivatives

[0193] The following sections describe two approaches that can effectively work to practically price quantum derivatives: Riemann summation and the reparameterization method of the subject disclosure. Riemann summation has been previously introduced, and a first resource analysis, the application of which achieves quantum supremacy, is described herein in accordance with one or more aspects of the subject disclosure. This analysis reveals the limitations of error scaling due to normalization. As described below, a new reparameterization method, which can be defined and / or implemented by the quantum resource estimation system 102 in accordance with one or more aspects of the subject disclosure, avoids the shortcomings of other methods and, in fact, provides the first end-to-end path to quantum supremacy.

[0194] 3.1 Riemann summation

[0195] The Riemann summation method provides an approach to construct the P-path loading operator in Algorithm 2.1. ndT Let be the size of the Hilbert space containing all possible paths.

[0196]

number

[0197] is the maximum value of the d asset multivariate transition probability in equation (2).

[0198]

number

[0199] teeth,

[0200]

number

[0201] and

[0202]

number

[0203] The normalized transition probability for all choices of the asset price in the interval [0,S max The steps of the method summarized in Algorithm 3.1 defined below are

[0204]

number

[0205] Using

[0206]

number

[0207] Calculate the price of the derivative as . Note that the normalization factor in the last step scales exponentially with T. P max < 1, no normalization is involved and this factor is redundant. However, P max >1, the error grows exponentially, making this approach impractical.

[0208] Algorithm 3.1 - Riemann summation pricing

[0209] Use n, d, and T, all of which are positive integers.

[0210]

number

[0211] The operator W applies the transition probability of the stochastic process to the ancillary process via t ,t=1,…,T.

[0212] 1. Apply Hadamard to the ndT qubit to prepare an equal superposition of all paths.

[0213] 2. Initial Price

[0214]

number

[0215] Load into the 0th qubit register.

[0216] 3.T transition operators W t Applying each of

[0217]

number

[0218] Configure.

[0219] where N=2 ndT is.

[0220] 4.

[0221]

number

[0222] is inserted into the quantum register and calculated,

[0223]

number

[0224] Get.

[0225] 5. Introducing ancillary qubits

[0226]

number

[0227] Rotate the value in the register to that amplitude.

[0228]

number

[0229] 6. Use amplitude estimation to extract the probability that the ancillary |1〉, which is the (e.g., discretized) expected payoff

[0230]

number

[0231] This can be rescaled to

[0232]

number

[0233] get.

[0234] Normalization factor P max is more tractable in return space, where the probability density function is given by equation (8). The logarithmic return is given by the asset volatility σ j When it is discretized to ±w times of

[0235]

number

[0236] is.

[0237] If the d assets are uncorrelated, then:

[0238]

number

[0239] Therefore, P max If ≦1,

[0240]

number

[0241] However, choosing a small discretization window w increases the truncation error explained in Section 2.3.1,

[0242]

number

[0243] in the case of,

[0244]

number

[0245] which grows linearly with the number of assets and time steps in the computation.

[0246] 3.1.1 Riemann summation error analysis

[0247] In addition to the truncation and discretization errors of Section 2.3, the Riemann summation approach contains errors due to scaling issues and quantum arithmetic.

[0248] When operating in return space, one transition operator is used to compute (12) and

[0249]

number

[0250] The transition operator performs amplitude encoding of the maximum summation error ε dens and the payoff operator for calculating equations (27) and (28) is the payoff error ε f Assuming that we introduce the total arithmetic error that can be estimated using amplitude estimation is:

[0251]

number

[0252] Neglecting the quadratic error term, we obtain:

[0253]

number

[0254] where the logarithmic return for each asset and each time step is given by the domain [-wσ max ,wσ max ] is configured to discretize

[0255] Probability density error ε dens is given by equation (12)

[0256]

number

[0257] have

[0258]

number

[0259] and arises from the calculation of the ancillary rotation in equation (26). The term in the exponential function in equation (12) is

[0260]

number

[0261] It can be written as:

[0262] where:

[0263]

number

[0264] and C ij are classical variables containing volatility and correlation parameters from the correlation matrix Σ. In equation (33),

[0265]

number

[0266] In each calculation of ε A There is an error of (d+d2)T times multiplication in total.

[0267]

number

[0268] The term is |w|σ by construction. max where each quantum register representing the logarithmic return R is bounded by a window [-wσ max ,wσ max Using the error analysis for addition and multiplication in Section 8.2, the total error in computing Equation (33) is

[0269]

number

[0270] is.

[0271] Then, using the error propagation analysis of Section 8.2, we compute the exponential, square root, arcsine, and sine functions on a quantum register that already contains arithmetic errors, and we get ε dens teeth,

[0272]

number

[0273] It is bounded by

[0274] Each rescaling performed on the input variables introduces a corresponding rescaling error. max In addition to rescaling, the quantum resource estimation system 102 scales the payoffs to 1 / f so that they lie in [0,1]. δ The final answer is

[0275]

number

[0276] These rescalings can be taken into account by multiplying by , and therefore the error in the estimate of the truncated integral by the amplitude estimate is

[0277]

number

[0278] The quantum resource estimation system 102 can then bound the error with a Riemann summation approach.

[0279]

number

[0280] where ε trunc , ε disc, and ε amp is defined as in Section 2.3.

[0281] 3.1.2 Resource Estimation

[0282] As an example, consider a basket autocallable with 5 autocall days and parameters T=20, d=3, and ε total / f δ ≦2×10 -3 The target error is set as follows. Select w~5 so that the truncation error of equation (20) is within the total target error, and equation (30) is P max ≒4 3 This gives a very large scaling factor,

[0283]

number

[0284] However, there may be several ways to address this normalization problem, such as the method inspired by importance sampling and discussed in Section 9.2. Therefore, assuming that some method for addressing normalization has been invented, P max =1.

[0285] Then, using the resource calculations discussed in Section 9.1, ε arith ≦2×10 -3 where p is the integer part of the fixed-point representation defined in equation (63). In this case, the Q operator uses 21k qubits and has a T-depth of 23k, including the resources to calculate the price from the logarithmic return using equation (11). Δt=1 / 20 and σ min If σ = 0.1, we calculate β ≒ 17. max = 0.4 and w = 5. Therefore, for the selection of n, ε disc ≒f δ 10 -5 and

[0286] ε trunc ≦f δ 10 -4 (37) is.

[0287] Amplitude estimation target ε amp to 10 -3 , the target reliability is α=10 -2 Then,

[0288]

number

[0289] This means that the total T depth is about 1.9×10 8 This means that

[0290] Using the same analysis, for a TARF contract with d=1, T=26, and Δt=1 / 26 (see, for example, Section 4.2), assuming an annualized volatility of the underlying asset of σ=0.4, ε total / f δ ≦2×10 -3 The target error is 1.7×10 8 A total T depth of 100 kJ and 15 k qubits can be achieved.

[0291] 3.2 Reparameterization Method

[0292] The limitations of normalization in Riemann summation motivate the development and implementation of new methods for loading stochastic processes. In a reparameterization method, the quantum resource estimation system 102 can move to modeling assets in return space. As described in Section 1.2, in return space, the underlying assets include uncorrelated normal distributions. It should be recognized that these different distributions can be loaded by preparing many standard normals (e.g., in parallel) and then applying an affine transformation to obtain the mean and standard deviation. This approach extracts a specific subroutine (loading the standard normal into the quantum state) and uses it as a resource to load the full distribution of the underlying path. The normal loading subroutine itself can be precomputed and optimized using variational methods. This is an advantageous combination of fault-tolerant quantum computing and variational compilation, which will be discussed in Section 3.2.1. Overall, the reparameterization method avoids the normalization problem in Riemann summation, reducing computational cost and / or resources. The steps in reparameterized pricing are described in Algorithm 3.2 defined below. In this context, the path ω R ∈Ω R is a series of logarithmic returns

[0293]

number

[0294] Note that the reparameterization method refers to P max Remove the problematic dependency on the operator G j can be implemented with relatively few resources using variationally learned circuits, as described in the following section.

[0295] Algorithm 3.2 - Reparameterized Pricing Method

[0296] Use n, d, and T, all of which are positive integers.

[0297] Standard Gaussian distribution

[0298]

number

[0299] We gain access to an operator G that loads an n-qubit register with g i , 2 n Let be the value of the probability mass function of a standard Gaussian distribution discretized into bins.

[0300] 1. Apply dT Gaussian operators G to ndT qubits. This constitutes:

[0301]

number

[0302] where:

[0303]

number

[0304] is the 2nd order of this multivariate standard Gaussian ndT Over all different realizations,

[0305]

number

[0306] denotes the corresponding probability.

[0307] 2.Σ=LL T Let be the Cholesky decomposition of the covariance matrix.

[0308]

number

[0309] to adjust the center and volatility of each Gaussian. The corresponding return paths and probabilities are denoted by ω R and p(ω R ) is shown.

[0310] 3. If the payoff can be calculated directly from the logarithmic return,

[0311]

number

[0312] quantum register

[0313]

number

[0314] and calculate directly.

[0315] First, if the payoff is defined in terms of price and not the proper logarithmic return,

[0316]

number

[0317] Calculate the price space path ω for each asset using: The calculation just described can be done in parallel for each asset (e.g., each derivative asset).

[0318] 4. Introducing an ancilla qubit and f(ω R ) register value to its amplitude as follows:

[0319]

number

[0320] 5. Use amplitude estimation to extract the probability that the ancillary |1〉, which is the (e.g., discretized) expected payoff

[0321]

number

[0322] This can be rescaled to

[0323]

number

[0324] get.

[0325] 3.2.1 Variationally Trained Gaussian Loader

[0326] The standard Gaussian loading operator G can be pre-computed since it is problem-independent in the reparameterization method. This section describes an approach to variationally optimize this operator. x _i =-w+iΔx, where i=0,…2 n -1, and Δx=2w / 2 n The standard normal distribution g(x _i ) in the example below. The domain has been fixed at w=5, so the total range of values ​​considered is 2w=10. This choice leaves approximately 5×10 outside the domain. -7 Finally, taking into account the different metrics used to normalize the function in real space and the wave function in the quantum register, it is normalized so that the sum of its squared elements is 1, and thus we aim to load the quantum register with the following distribution (for example, the target distribution):

[0327] g(x i )×Δx, Σ i g(x i )×Δx=1. (41)

[0328] Note that due to the finite truncation region, the target distribution is normalized to 1-α. In principle, the distribution could be renormalized to 1 in a chosen interval of width 2w. In any case, this choice provides a negligible difference when compared to the error observed in training.

[0329] The variational ansatz chosen is represented by the so-called Ry-controlled NOT (Ry-CNOT) ansatz, which has linear connectivity (see, e.g., Section 11.0). Various aspects of the subject disclosure provide a novel strategy for optimizing circuits in this context, which relies on energy-based methods and is also detailed in Section 11.0. In essence, the target cost function is the energy of the associated quantum harmonic oscillator problem, whose ground state is naturally Gaussian. Note that the discretized quantum harmonic oscillator solution (e.g., the squared coefficient of the solution) can match a normal distribution only in the limit Δx→0. To correct for this, we perform the following re-optimization, which directly targets the infinity norm between the two distributions.

[0330]

number

[0331] Here, the quantum state encoded in the register is given by the coefficient

[0332]

number

[0333] is defined by

[0334] Note that directly learning Equation (42) as the cost function is inefficient, so pre-training using an energy-based approach is used. ∞We observe how the cost function exhibits a much more waveform landscape in the circuit parameter space compared to the energy of the related quantum harmonic oscillator problem.

[0335] Note that for different choices of register size n, the circuits encoding these Gaussian states can be pre-trained in advance, so training is not included as part of the run time for any given derivatives pricing problem. Figure 4 shows results for different register sizes and circuit Ansatz depths. More details are provided in Section 11.0.

[0336] 4 illustrates an exemplary, non-limiting graph 400 that can facilitate estimation of quantum resources for calculating the expectation value of a stochastic process using a reparameterization method according to one or more embodiments described herein. Repetitive descriptions of similar elements and / or processes utilized in each embodiment have been omitted for the sake of brevity.

[0337] Graph 400 illustrates the L1 approximation by training a variational Ry-CNOT circuit (e.g., via quantum resource estimation system 102 and / or variational component 202) to approximate G for different register sizes n. ∞ error. This numerical study shows that states that can be variationally prepared approach the target exponentially faster in the depth and hence the number of gate operations. This observation is in good agreement with the behavior expected from the Solovay-Kitaev theorem, which gives an upper bound on the number of gates that can be used to achieve a desired accuracy of the cost function. Indeed, for any target operation U∈SU(2 n ) for SU(2 n )

[0338]

number

[0339] exists, and the energy error ε is the depth D=O(log c (1 / ε)). The entanglement block in the circuit causes the SU(2 n ) operations are a subset of SU(2 n ), it can be numerically observed that the exponential reduction in error with the number of gates still holds.

[0340] At the end of this section, we investigate the portability of these results in a fault-tolerant regime, which could enable the application of entire derivatives pricing algorithms. While the numerical results provide evidence of a fairly efficient Gaussian state preparation in terms of the circuit depth of Ry-CNOT, additional steps are taken in terms of fault-tolerant implementation of such circuits. In this new framework, the continuous rotation Ry gate is extended as a finite product of discrete operations. Again following the Solov'y-Kitaev theorem or a more specialized result

[27] , it is also possible to obtain an efficient representation of any SU(2) operator as a sequence of Clifford+T gates that scales logarithmically with the threshold error ε. We investigate how previously obtained results can be transferred in this regime, where the angle of rotation can only take discretized values. Thus, for each parameter

[0341]

number

[0342] is in the format j*2π / M digit where j is an integer. Section 11.0 explains that the error introduced by such digitization is 1 / M digit We numerically show how the σ decreases systematically with mesh size.

[0343] 3.2.2 Error analysis

[0344] The total error in the reparameterization approach is:

[0345]

number

[0346] where ε trunc , ε disc , and ε amp are the truncation, discretization, and amplitude estimation errors bounded in Section 2.3. Here, the term ε arith arises from the individual errors introduced during the preparation of the Gaussians and the calculation of the payoff. i ) is L ∞ Error ε dens and the payoff calculation is ε f Assuming that this results in a maximum error of , the total error is:

[0347]

number

[0348] where x=(x1,x2,…,x dT ) Expanding the integrand and keeping only the linear error term gives:

[0349] ε arith ≦2wdTf δ ε dens +ε f , (45)

[0350] where:

[0351]

number

[0352] is used by truncating the probability mass function.

[0353] 3.2.3 Resource Estimation

[0354] Calculate the resources that can be required for the same basket autocallable as in Section 3.1.2, where d = 3, T = 20, Δt = 1 / 20, σ max =0.4, σ min = 0.1, and w = 5, and the contract has 5 autocall days. Further, assume that each Gaussian is prepared with n = 5 qubits, ε dens =2×10 -6 , ε amp =ε f =10 -4 So that, ε total / f δ ≒2×10 -3 From Figure 4, we use 5 qubits and a circuit depth of 6 to obtain the total error of L ∞ ~2×10 -6 It can be seen that a Gaussian state can be prepared with α = 10, requiring 7 layers of Ry gates. Using these inputs and the resource calculations described in Section 10.0, constructing the Q operator using reparameterization requires 7.5k qubits and has a T-depth of 5.7k, which includes the calculation of the price from the logarithmic return, i.e., Eq. (11). -2 At the target confidence level of , the total T depth is 4.6 × 10 7 Using the reparameterization method, ε total / f δ ≒2×10 -3 To price the TARF in section 3.1.2 with d=1, T=26, Δt=1 / 26, and σ=0.4 with an accuracy of 6×10 7 It uses a total T-depth of and 9.5k qubits.

[0355] 4.0 Payoff

[0356] 4.1 Autocallable Contracts

[0357] Typically, an autocallable contract is defined in terms of asset returns compared to a predefined reference level and includes a notional amount that is used to calculate the dollar value of the contract. In the case of a single underlying asset, an autocallable can include:

[0358] Strike (strike; exercise of rights) K i , exercise time t i , fixed payoff f i A set of binary options {(K i ,t i ,f i )} i=0…m-1 These are t i <t i+1 Assume that it is sorted so that

[0359] Strike K o and short knock input with barrier b, and

[0360] If any binary option pays off, all subsequent options and puts are knocked out.

[0361] The strike and barrier parameters are defined in terms of the return of the underlying asset price S(t) compared to a reference level, which, without loss of generality, can be taken as the initial spot price S0 of the underlying asset. The payoff f of a binary option i is similarly defined as a dimensionless parameter indicating return. In a return space where the basis vectors represent the logarithmic returns of the underlying asset (see Section 1.2), checking whether the underlying asset has crossed a strike or barrier K given a logarithmic return value R is done by using e R It may include checking whether ≧K.

[0362]

number

[0363] Let be the normalized payoff given by:

[0364]

number

[0365] Algorithm 4.1 – Autocallable Payoff Implementation

[0366] Parameter {(K i ,t i ,f i )} i=0…m-1 ,K o Find an autocallable with a and b.

[0367] 1. For each time step t = 1...T, the cumulative return calculated by the selected path distribution loading method.

[0368]

number

[0369] Assume access to

[0370] 2.Each t i Apply a comparator in parallel to

[0371]

number

[0372] get.

[0373] 3.

[0374]

number

[0375] Successively, for each bit in the strike register, θ i We apply a controlled rotation of x to the accumulator qubit, which is shown in Figure 5. This introduces an m-qubit ancillary register a.

[0376] 4. Each cumulative return

[0377]

number

[0378] Applying a comparator in parallel to the register

[0379]

number

[0380] to indicate whether the put option was knocked in and whether it is in the money. These bits are then ORed together to obtain |put>1, which holds whether the payoff from the put option is being considered.

[0381] 5.

[0382]

number

[0383] and normalize it using equation (46) to obtain the put option payoff |f p 〉 is required.

[0384]

number

[0385] Calculate.

[0386] 6. Then, controlled R y Using rotation to control both |put〉 and a[m-1]

[0387]

number

[0388] rotates bitwise to the target qubit.

[0389] 5 illustrates an exemplary, non-limiting circuit 500 that can facilitate estimation of quantum resources for calculating the expectation value of a stochastic process using a reparameterization method according to one or more embodiments described herein. Repeated descriptions of similar elements and / or processes utilized in each embodiment are omitted for the sake of brevity.

[0390] Circuit 500 comprises an exemplary, non-limiting circuit that can be used to accumulate binary option payoffs in an autocallable with five binary options, where qubits s,...,s represent five strike values ​​K i Represents a Boolean comparison of the payoff f i (e.g., given by a particular phase of the R-Y rotation) occurs only if no payoff occurred previously. The entire payoff is loaded into the amplitude of qubit e.

[0391] The amplitude estimate is the expected return of the contract.

[0392]

number

[0393] Its dollar value is

[0394]

number

[0395] where N is the notional amount specified in the contract. Autocallables can be defined not only on one asset but also on a basket of assets. Typical examples include BestOf and WorstOf, where the contract's return is based on the return of the best or worst performing asset in the basket. These may be treated similarly to the single asset case, in which case the return of each asset is

[0396]

number

[0397] are first compared to find the maximum or minimum (e.g., as appropriate).

[0398] Steps 2 and 4 of Algorithm 4.1 defined above can be implemented using logic circuits (e.g., comparators, ANDs, ORs) that do not introduce errors, while steps 3 and 6 involve the use of a controlled Ry rotation whose decomposition into T gates is a function of the summation error ε, which can be selected depending on the desired precision of the calculation. Step 5 is the most resource-heavy component of the payoff circuit, which is the quantum register

[0399]

number

[0400] This can include computing,the,divide of that register by a classical constant in the denominator of,Equation (46), and computing the square root and arc sine of the,register.,The resource criteria for all the above circuit components are,detailed in Section 8.1, and the corresponding arithmetic and,gate synthesis errors are discussed in Section 8.2.

[0401] Consider again the autocallable contract in Section 3.1.2 and Section 3.2.3, which has 5 autocall dates, is defined on d = 3 assets, and is simulated using T = 32 time steps. The total additive payoff error ε determines the resources that can be used by each component when distributed across the operations in steps 3, 5, and 6 of Algorithm 4.1. f Target ε f =10 -4 For,assuming that the computation can be parallelized as much as possible, the circuit,that computes the autocallable payoff contains 1.6k qubits, and a,T,depth of 2k.

[0402] 4.2 TARF

[0403] This section describes the TARF implementation for a single underlying asset in the price space.

[0404] TARF is as follows:

[0405] Forward price F, payment dates t1,…,t m , two strike prices K upper and K. lower ≦F, knockout price K o , and accrual cap

[0406]

number

[0407] The date is t i ≦t i+1 Assume that it is sorted so that

[0408] ·Each time t i In , TARF has payoffs as follows:

[0409]

number

[0410] Any t i When the price is K o > F, all subsequent payoffs are knocked out.

[0411] · An accrual cap condition that if the total profits accumulated by any payment date exceed C, the contract holder will receive only those payoffs that bring the total profits equal to C, and the remaining forward contracts will be knocked out.

[0412]

number

[0413] Let be the normalized payoff given by:

[0414]

number

[0415] Algorithm 4.2 – TARF Payoff Implementation

[0416] Parameters (F, t1, ..., t m ,K upper ,K lower ,K o , C) is obtained.

[0417] 1. Knockout and accrual cap qubits |0〉 o , |0〉 c Start from.

[0418] 2. All time t i Apply three comparators in parallel to

[0419]

number

[0420] get.

[0421] 3. Each time t i Regarding

[0422] (a)

[0423]

number

[0424] |·〉 o and ORed onto the new qubit, which is | 〉 o It can be relabeled with

[0425] (b) Compute the payoff (e.g., conditional on different strike qubits) and add the payoff to a register that records the total payoff,

[0426]

number

[0427] get.

[0428] (c) Apply a comparator to

[0429]

number

[0430] Calculate whether the accrual cap has been reached to get

[0431] (d)

[0432]

number

[0433] Calculate the quantity that makes the total equal to C to get

[0434] (e) Normalized payoffs for reaching and not reaching the accrual cap

[0435]

number

[0436] , and calculate the AND of each condition, (for example, a new) knockout qubit | 〉 o NOT, accrual cap qubit |·〉 c In addition, the appropriate one controlled by NOT

[0437]

number

[0438] get.

[0439] (f)

[0440]

number

[0441] and the cap qubit | 〉 c and store the result in a new qubit, which is | 〉 c It can be relabeled with

[0442] 4. |θ〉

[0443]

number

[0444] Calculate

[0445] 5. Final encoding qubit |0〉e Above, for each qubit i in |θ〉, the controlled angle 2 -i A series of controlled R y Apply rotation.

[0446] 5.0 Discussion

[0447] Various aspects of the subject disclosure described herein provide complete resources and error analysis for pricing financial derivatives using quantum computers. In particular, these various aspects use autocallables and TARFs, two types of path-dependent options that are relevant in practice and difficult to price classically, as illustrative case studies. To this extent, various aspects of the subject disclosure provide new methods for loading stochastic processes that overcome limitations of existing approaches. While these various aspects involve geometric Brownian motion, the subject disclosure is not so limited, and the approaches described herein can be readily extended to stochastic or local volatility methods, for example, by loading multiple independent stochastic processes and having conditional or non-stationary reparameterizations.

[0448] The resource estimates presented herein provide a target performance threshold for quantum computers that can be advantageous in derivatives pricing. Assuming a 1-second target for pricing autocallable options, 10 1 The code distance that can support one logical operation will have a logical clock rate of 10 MHz.

[0449] Current estimates of logical clock rates are around 10 kilohertz (kHz), or roughly three orders of magnitude below, but future research into algorithms, circuit optimization, error correction, and hardware will continue to improve resource metrics and execution times. For example, in the case of Shor's algorithm, careful analysis across several publications has reduced the estimated resource metrics by nearly three orders of magnitude. The subject disclosures described herein represent a first milestone on the journey toward quantum advantage for pricing financial derivatives.

[0450] 6.0 Background on Derivatives

[0451] 6.1 Forward

[0452] An example of a derivative is a forward contract (often simply called a forward). In a forward, the holder promises to buy or sell a particular asset to the issuer at a fixed price F, known as the forward price, on a specified date in the future. A simple path-independent example is when the holder promises to buy a quantity x of an asset m months from now at $F per asset. Forwards are typically cash-settled, meaning that rather than an exchange of money for an asset at maturity, there is only an exchange of money determined by the value of the asset, with a payoff determined by this payoff. For example, if the price of an asset at maturity date T is S, then T If , the payoff is f(S T )=x(S T -F), and f(S T )>0, the contractor makes a profit (the issuer makes a loss), and f(S T )<0, the opposite is true.

[0453] 6.2 Options

[0454] Another example of a derivative is an option. An option can be thought of as a conditional forward. Unlike a forward, where the issuer is obligated to buy or sell an asset, in an option contract the holder has the option to buy or sell a specific asset from the issuer at a predetermined price at a future date. If the holder chooses to buy or sell the asset, they have chosen to exercise the option. Like forward contracts, option contracts are typically settled in cash based on the value of the asset on the exercise date. An example of a path-independent option with a single underlying asset is the European call option, where the issuer has the option to buy the asset at a strike price K on the expiration date. The payoff at expiration is f(S T )=max(S T -K,0). In a European put option, the issuer has the option to sell an asset at strike price K on the expiration date, and f(S T )=max(KS T , 0). Another example of a path-independent option is a binary option with a fixed payoff if the underlying asset is above or below the strike at time T.

[0455] 6.3 Path Dependence and Discounted Payoffs

[0456] An example of a path-dependent derivative is the knockout European call option, which is the same as a European call option but with an additional knockout price, π. If the underlying asset rises above this value at any time between 0 and T, the contract becomes worthless. This path-dependent payoff function has the following form:

[0457]

number

[0458] Including the value of the underlying asset at times other than T introduces path dependence. Another example is a knock-input option with a payoff such as:

[0459]

number

[0460] Here, the contract is a knock-in because there is a non-zero payoff only if the asset falls below a value π, the amount by which it falls below the strike price K.

[0461] The examples discussed so far have only one payment date at time T, when the exchange occurs between the contract issuer and holder. Some path-dependent options can have several payment dates, where multiple payments occur at different times throughout the contract's life.

[0462] We now introduce the concept of discounted payoff. As expected, the current price of a derivative is related to its expected payoff in the future. However, a time lag of the payoff can also be considered to account for the opportunity cost of investing in a risk-free asset at interest rate r. If a contract is made between now and time t i Payoff at f i , the discounted payoff can be defined as:

[0463]

number

[0464] The price of a derivative contract is given by the expected value of the discounted payoff in the underlying stochastic process. In practice, path-dependent derivatives are much more difficult to price computationally and are often priced using Monte Carlo simulations of paths. This contrasts with models for path-independent derivatives, which may have analytical solutions, such as the Black-Scholes model for European call options. Path-dependent options offer an opportunity to take advantage of quantum speedups to Monte Carlo.

[0465] 6.4 Autocallable Options

[0466] A typical example of an autocallable option is a set of binary options, each of which pays a different fixed payout on a different payment date, and which knocks out the entire instrument (e.g., voids all future payoffs) if it pays out on any of the payment dates. More formally, (K,t i ,f i ) at time t i If the underlying asset value is above the strike price K, the payoff f i where the autocallables are the set:

[0467] {(K,t1,f1),(K,t2,f2),…,(K,T,f T )}, (52)

[0468] where {t i} and {f i} typically grows linearly. i ,f i ) pays out (in the money), all subsequent options {(K,t j ,f j )} j>i is knocked out (e.g., disabled).

[0469] In practice, these binary options are often bundled with short knock-in options, i.e., knock-in options granted to the issuer by the holder, thereby reducing the risk for the issuer and lowering the price for the client. As with the set of binary options, this put option can also be bundled with a binary option (K,t j ,f j ) will be knocked out if any of them pay out.

[0470] As is common with many options, autocallables can be extended to have multiple underlying assets. In this case, it is typical to tie the entire option payoff to the best- or worst-performing asset, where performance is defined in terms of return, and the knock-in option payoff is proportional to the return of the best- or worst-performing asset if that asset is below the strike price. The strike for the underlying j is often the same in the return space, i.e., for each asset K j The strike price is

[0471]

number

[0472] Note that k is the same constant for all underlying assets. Thus, if the worst-performing asset is in-the-money (e.g., above the strike price), then so are all other assets. Conversely, if the best-performing asset is below the strike price (or out-of-the-money), then so are all other assets. In principle, different underlying assets could have independent strike prices, but this is not common, so all strike prices are

[0473]

number

[0474] An assumption can be made that is defined as:

[0475] Contingent payoffs, and knock-inputs, mean that autocallables have strongly path-dependent payoffs, which means that they are computationally expensive in practice, taking sometimes 5-10 seconds using classical Monte Carlo methods with at least 40,000 paths.

[0476] 6.5 Target Accrual Redemption Forward

[0477] A target accrual redemption note (TARN) is a derivative with a payoff capped at a specific target amount. Historically, this term referred only to bills of exchange, hence its name, but now it encompasses any derivative with an accrual cap. Various aspects of the disclosure being targeted use a commonly employed TARN called a target accrual redemption forward (TARF) as an exemplary case for implementing the quantum resource estimation system 102. A TARF is a forward set with several knockout conditions. Specifically, it is a derivative with a single underlying asset having several (e.g., 20 - 60) payment dates and a forward price F. Throughout the contract, K upper =F and there can be two fixed strike prices K lower <F. At each payment date t i there are several possible payoffs as follows:

[0478]

Number

[0479] Here,

[0480]

Number

[0481] is the price of the underlying asset at payment date t i and α is a positive constant. <00​​​​​​​​​Note that when , the payoff is negative and the holder of the derivative suffers a loss. The constant α makes this loss asymmetric, and when it occurs, it is often 1 or 2.

[0484] Furthermore, TARF has two knockout conditions based on the knockout threshold π and the accrual cap C. The first condition states that if the price of the underlying asset is equal to or greater than π on any payment date, the derivative contract will be knocked out immediately (e.g., without payment on that day). The second condition states that if the price of the underlying asset is equal to or greater than π on any payment date t i The total payoff for the holders is the payoff f i would exceed the accrual cap C, the contract holders would instead receive only the amount that would bring their total payout to C, and the contract would be knocked out.

[0485] 7.0 Disadvantages of Global Dollar Loading

[0486] The Groß-Rudolph algorithm is often cited as an efficient way to create quantum superpositions corresponding to classical distributions. Given a probability distribution {p i}, the algorithm creates a quantum superposition of the form:

[0487]

number

[0488] The algorithm is inductive in nature and first assigns a probability distribution to a number of variables in the domain of interest. m We start by assuming that there is a way to divide the world into regions and create the following states:

[0489]

number

[0490] where:

[0491]

number

[0492] is the probability that the random variable is in region i. Then, add one qubit to the state of equation (55) and m The region is further divided into two probability distributions with evolution of the following form: m+1 Aim to discretize:

[0493]

number

[0494] where α i (β i ) is the probability that the random variable is in the left or right half of region i.

[0495]

number

[0496] and

[0497]

number

[0498] Let be the left and right boundaries of region i, then the function

[0499]

number

[0500] is the probability that x is also in the left half of region i, given that it is in region i. A circuit can be constructed that can perform the following calculations:

[0501]

number

[0502] where:

[0503]

number

[0504] Then, the angle θ at the m+1th quantum bit i The controlled rotation of

[0505]

number

[0506] |θ i After uncompute 〉,

[0507]

number

[0508] which is the extension of the state of equation (55) to one additional qubit. Performing this iteration n = log2N times gives a discretization of the distribution over a total of N points across the n qubits.

[0509] In practice, the efficiency of the Grobel-Rudolph method depends on the ability to perform the integration of equation (57) in superposition. The argument in the original formulation is that probability distributions that can be efficiently integrated classically using probabilistic methods (e.g., using Monte Carlo) can equivalently be efficiently integrated quantum-wise. However, since the ultimate goal of quantum derivatives pricing is to provide a faster alternative to Monte Carlo integration for probability distributions, performing this integration as part of initial state preparation without a corresponding quantum speedup negates the advantages offered by amplitude estimation as an alternative to Monte Carlo. While efficient from a complexity perspective, this means that Grobel-Rudolph is an insufficient method for quantum supremacy in derivatives pricing.

[0510] More recently, an approximate method for implementing the Grobber-Rudolph algorithm for the standard normal probability distribution has been presented, and the authors propose a formula for equation (57) written as follows:

[0511]

number

[0512] This can be approximated as follows:

[0513]

number

[0514] The authors emphasize that for small δ, the approximation in (62) becomes sufficiently accurate for m ≥ 7 as the δ parameter decreases with each iteration of the Großer-Russell algorithm that adds qubits to the discretization. However, because the Großer-Russell construction is iterative, it is possible for terms for m < 7 to be calculated before the above approximation is possible. Therefore, the integral in (57) is calculated classically and loaded into the corresponding quantum register. While this approximation allows for a simplification of the general Großer-Russell algorithm for the standard normal distribution after a certain point in the iteration, it does not change the fact that it may involve computing the integral over the entire domain of the probability distribution, and therefore becomes practically infeasible for the same reasons as the original Großer-Russell method.

[0515] 8.0 Fixed-Point Quantum Arithmetic Resources

[0516] This section describes general quantum arithmetic operations and provisions for the composition of arbitrary rotations. These operations are used for resource estimation and error analysis. Quantum arithmetic can be included in path loading using the Riemann sum method (Section 3.1), the reparameterization method (Section 3.2), and the payoff calculation described in Section 4.0. All arithmetic operations included in Equation (12) can be performed using the Riemann sum method, as well as the calculation of the arcsine and square root of quantum registers for the payoff calculation in Equation (15). Efficient algorithms for individual arithmetic operations have been identified, and resources are typically reported as a number of Toffoli gates or T-gates. When arithmetic algorithms are implemented from prior work in the literature, gate costs are reported herein for the gate set reported by the authors.

[0517] In a fault-tolerant setting, an estimate of the T-depth of the circuit in the Clifford + T-gate set decomposition can be made, and an assumption can be made that a Toffoli gate can be constructed with a T-depth of 1 using ancilla qubits. For each operation, an assumption can be made that the resulting circuit can be parallelized as much as possible.

[0518] 8.1 Resource Estimation

[0519] All calculations are done in fixed-point arithmetic. The n-bit representation of a number x is:

[0520]

number

[0521] where x i ∈0,1 denotes the i-th bit of the binary representation of x, and p denotes the number of bits to the left of the binary point. The selection of n and p controls the error that can be tolerated in each calculation and the resources that can be used to perform arithmetic operations on registers. Once the values ​​of (n,p) are chosen so that the total arithmetic error is acceptable for the problem under consideration, keep them constant throughout the analysis. It is possible to adjust these values ​​for different components of the circuit to reduce the total resources that can be used, but for simplicity of explanation of the subject disclosure, this potential optimization can be ignored.

[0522] TF f and T f Let n denote the number of Toffoli gates and T-depth used to compute an arithmetic function or logical operation f. The estimate of the operations is a function of the fixed-point register size (n, p) used to represent the underlying quantum states involved in the computation.

[0523] Addition and / or subtraction

[0524] Addition of two given n-qubit registers is performed with a Toffoli cost of 2n-1. Subtraction is given by ab = ~(~a + b), so it can be implemented as an addition with 2n additional X gates, which does not change the Toffoli count.

[0525] Given the T-depth cost of controlled and uncontrolled addition, the addition circuit is T-depth regardless of register size. add = 10, and controlled addition can be constructed with a T-depth cost of O(n) for registers of size n.

[0526] Multiplication

[0527] Multiplication uses a controlled adder circuit and the following Toffoli counts:

[0528]

number

[0529] This method may also be used to divide a quantum register by a classical value, which can be done by inverting the classical value and utilizing a multiplication algorithm.

[0530] Fixed-point multiplication involves n controlled additions, and is therefore O(n 2 ) has a T-depth cost. These methods can use ancillary qubits proportional to the register size, but because the circuit involves inversely computing the ancillary qubits, the ancillary qubits can be reused for each subsequent addition, which is not done in parallel. Because the computation can be parallelized over d assets and T time steps, include an additional T*d*n qubits when counting the sum to account for these potential ancillary qubits.

[0531] Furthermore, by considering the registers of a factor of 1 as z≧1 independent registers of size n / z, each multiplication circuit can be parallelized and each controlled addition can be performed in parallel on z sub-registers. This allows the z sub-results to be accumulated into a final result using n·(z-1) additional qubits and z-1 additions. z=1 indicates that no additional parallelism is employed. If the pairwise accumulation additions can also be parallelized, then the total T-depth cost of a parallelized fixed-point multiplication is given by:

[0532]

number

[0533] square root

[0534] We utilize the square root algorithm, which can be extended for quantum registers in fixed-point representation: for a number x of size (n,p), we compute the square root by treating x as an n-digit integer and shifting the result to the right (np) / 2 times.

[0535]

number

[0536] This amounts to:

[0537]

number

[0538] The Toffoli count for this square root algorithm is:

[0539]

number

[0540] As reported by the authors, the T-depth of this algorithm is T sq(n)=5n+3, so 2n+1 qubits can be used.

[0541] Logical operations

[0542] For comparisons between quantum registers, or between a quantum register and a constant,

[0543]

number

[0544] A logarithmic comparator with a Toffoli / T depth of 1 is used, which includes an intermediate ancillary element. A logical OR operation on a two-qubit input can be performed with a Toffoli / T depth of 1.

[0545] Exponential Function

[0546] A general quantum algorithm can be used to compute smooth classical functions using parallel piecewise polynomial approximations. It is applied to estimate the resources to compute exponential functions. The algorithm takes parameters k and M, which control the polynomial order and the number of domain subintervals selected for the piecewise approximation, respectively. The Toffoli sum is given by:

[0547]

number

[0548] This algorithm can also be used to calculate the arcsine function, which involves k iterations of multiplication and addition, using a k-th order polynomial for the approximation. Furthermore, for the M selected subintervals, it includes M comparison circuits between an n-qubit input register and a classical value.

[0549]

number

[0550] Using a comparator with a T-depth of , the T-depth of the parallel polynomial evaluator is:

[0551]

number

[0552] where z is the optional parallelization factor introduced in the above resource estimation for the multiplication circuit.

[0553] The number of qubits in the parallel polynomial evaluation scheme for a choice of polynomial order k and number of subintervals M is:

[0554]

number

[0555] arcsine

[0556] To calculate the arc sine, we utilize the general quantum algorithm described above, just as can be done for exponential functions. However, the derivative of:

[0557]

number

[0558] diverges around ±1, so the authors of the general quantum algorithm use the following transformation:

[0559]

number

[0560] to handle the interval x∈[0.5,1]. Since the computation of the arc sine may involve a conditional square root evaluation of the argument, whenever the arc sine is computed we also compute the square root (e.g., equation (15)) and use the following transformation:

[0561]

number

[0562] The resource estimation considerations are similar to those in the prior art approach.

[0563] Indicates whether the above transformations should be applied,

[0564]

number

[0565] Assuming the values ​​in the quantum register are normalized, two Toffoli gates can be used.

[0566] ·

[0567]

number

[0568] or

[0569]

number

[0570] Conditional subtraction and conditional copy are performed depending on the value of the comparator above to prepare either of the following. The conditional copy can use n Toffoli, and the conditional subtraction is performed using TF add +n Toffoli gates can be used.

[0571] TF for square root calculation sq Toffoli gates.

[0572] Toffoli gate used in polynomial evaluation to calculate the arc sine

[0573] When x<0.25

[0574]

number

[0575] , otherwise

[0576]

number

[0577] A conditional copy and conditional subtraction that again depends on the result of the comparator from the first step to get

[0578] Using these considerations and the Toffoli count of the polynomial approximation of arcsin(x),

[0579]

number

[0580] The total Toffoli count to calculate is:

[0581]

number

[0582] of a number x represented by a register of size (n,p), calculated similarly to the exponential function

[0583]

number

[0584] The T depth for calculating is:

[0585] T arcsq (n,p,z)=T sq (n)+T pp (n,z)+8n+6, (75)

[0586] where T sq (n)=5n+3 is the T-depth for the square root algorithm.

[0587] This operation is q arcsq qubits, where the qubit reference for the arcsine may be given by equation (70) for choices of k and M and 2n+1 for the square root operation as follows:

[0588] q arcsq (n,k,M)=q pp (n,k,M)+2n+1 (76)

[0589] R y

[0590] R in the variational preparation of the Gaussian discussed in Section 3.2.1 y (θ) rotation, and a controlled R to encode the payoff into the ancillary amplitude in Eq. (16). y It uses rotations, as well as transition probabilities in the Riemann summation method. Using the Repeat-Until-Success method, any single-qubit unitary can be performed to within precision ε with a T-depth of approximately 1.15 log2(1 / ε) using one ancilla qubit and measurements.

[0591] When the rotation angle θ is stored in a separate register |θ〉, a series of R y (θ k ) rotations, each controlled by the kth qubit in |θ〉, where

[0592]

number

[0593] is.

[0594] A single controlled R n is a single R n depth, three Rs n count, and a single ancilla qubit, and can be executed using decomposition. However, since each rotation contributes an error ε, if |θ〉 is an n-qubit register (e.g., having p bits to the left of the binary point), the end-to-end operation can be executed with a precision of ε at a maximum T depth of 1.15n log2(n / ε). θ k A controlled R with <arcsin(ε)]] n Note that the amplitude increase due to the rotation is smaller than ε and thus unnecessary, and reduce this depth slightly. Thus, using this observation and Equation (77),

[0595]

Number

[0596] Calculate the total number of rotations that result in. This gives the final T depth of the controlled Ry(θ) operation as follows:

[0597]

Number

[0598] where

[0599] <'

Number

[0600] is.

[0601] 8.2 Error analysis

[0602] Given the fixed-point representation of equation (63), each arithmetic operation requiring a register will result in some approximation error depending on the particular method used. Here we outline the arithmetic errors associated with each operation described in the previous section.

[0603] Addition and / or multiplication

[0604] It uses fixed-point addition and multiplication and adds two (n,p) size registers.

[0605]

number

[0606] If the error bounded by , then the error associated with the multiplication is at most:

[0607]

number

[0608] For (n,p) size registers X and Y, each register has an additive error ε X , ε Y If we already include the above and each factor X and Y is bounded by b, the error in the calculation of X and Y is given by:

[0609] ε mul =b*(ε X +ε Y )+ε X ε Y +ε M (n,p) (80)

[0610] Exponential Function

[0611] Parallel polynomial evaluation methods are used to estimate the resources and associated errors in computing exponential functions. The error associated with the algorithm depends on the degree of the selected polynomial approximation and the choice of the number of subintervals, but the authors of such algorithms have -5 From 10 -9 We provide explicit error estimates and corresponding resources for errors in the range ξ, ... exp Showing that the total arithmetic error in computing the exponential function of a register can be approximated to first order in the Taylor expansion of exp(-x+ξ) as follows:

[0612]

number

[0613] square root

[0614] As discussed in the previous section, for square root calculations, consider the square root algorithm extended to quantum registers in fixed-point representation. The mapping in equation (66) can result in a maximum error of

[0615]

number

[0616] When computing the square root of a register x that already contains a (say positive) additive error ξ, the total additive error due to the square root operation is

[0617]

number

[0618] This is the estimated value

[0619]

number

[0620] This can be easily understood by observing that using the square root algorithm gives

[0621]

number

[0622] and then

[0623]

number

[0624]

number

[0625] where the first inequality is, for positive x and ξ,

[0626]

number

[0627] is derived from.

[0628] arcsine

[0629] For the arcsine calculation, we again used the polynomial estimation method, and the sample resource estimate of the error rate was 10 -5 From 10 -9. We first bound the error from computing the arcsine on a register, including the arithmetic error ξ. As discussed in Section 8.1, we compute arcsin(x) for x≦0.5. Furthermore, when the algorithm of the subject disclosure described herein computes the function arcsin(x), we do so only for x≧0. This provides a domain for the arcsin(x) error computation: 0≦x≦0.5. Note that given this domain, the slope of arcsin(x) is always monotonically increasing, with a maximum at x=0.5. Therefore, computing the error at x=0.5 gives us the following upper bound:

[0630]

number

[0631] where ε arcsinは , is the error due to the calculation of the arc sine given the choice of polynomial order and number of subintervals.

[0632] sine

[0633] As discussed in the previous section, we compute the sin(θ) function with a series of controlled Ry rotations controlled by qubits from a register containing the angle θ. We bound the error from computing sin(θ) when the register assumed to represent θ actually represents θ + ξ due to arithmetic error. To quantify the upper bound, we note that in the region 0 ≤ θ ≤ π / 2, the slope of sin(θ) is monotonically decreasing, and therefore has its maximum slope at θ = 0. Therefore, computing the error when θ = 0 yields the following upper bound:

[0634]

number

[0635] Here, the inequality sin(a+b)≦sin(a)+b for b≧0 is used, and ε sinis the error resulting from the gate decomposition of the Ry operator described in Section 8.1.

[0636] 9.0 Riemann summation

[0637] 9.1 Riemann sum path loading resource estimation

[0638] In this section, we consider the depth and number of qubits in a quantum register to calculate Equation (12) and encode the value into the amplitude of the ancillary qubits as described in Algorithm 3.1. The calculation is performed in logarithmic return space (see Section 1.2) and includes resource estimates for the operations described in Section 8.1.

[0639] T f and q f Let denote the T-depth and the number of qubits of operation f, respectively. Assuming that the computation over d assets and T time steps can be parallelized as much as possible, we obtain

[0640]

number

[0641] have

[0642]

number

[0643] The resource contribution to compute

[0644] T to calculate the term (R-μ) which can be done in parallel for d assets and T time steps add Here, T*d*n qubits are used to hold the logarithmic return R for s assets and time steps.

[0645] All R in the expansion of Eq. (33) (e.g., in parallel for all d and T) that acquire and / or use T*d*n additional qubits 2 T for the term mul .

[0646] All R in the expansion of (33) (e.g., parallel with respect to d,T) i R j T for the term mul *d2 / (d / 2) and T*d2*n qubits.

[0647] To sum all terms in equation (33) in parallel

[0648]

number

[0649] The qubits from the previous step can be reused here.

[0650] · q given by equation (70) for the choice of parameter values ​​determined by the desired approximation accuracy exp With q exp T to calculate the exponential function in equation (8) using an additional qubit exp .

[0651] The qubit resource is given by equation (76),

[0652]

number

[0653] T to calculate the arc sine and square root of arcsq .

[0654] ·All sums for t'∈[2,T] in Eq.(11)

[0655]

number

[0656] T to calculate add *(T-1) and (T-1)*d*n qubits

[0657] ·q exp T to compute prices across all assets and all time steps in equation (11) in parallel using *d*T additional qubits. exp .

[0658] ·

[0659]

number

[0660] T-depth of 1.15nlog2(n / ε) for an ancillary rotation to precision ε, controlled on the register where R is calculated. y It contains n ancilla qubits using decomposition.

[0661] Furthermore, when using the parallel multiplication scheme described in Section 8.1 to compute prices across assets and time steps, additional registers of size T*d*n are included to implement a summation circuit with a constant T depth and (z-1)*T*d additional qubits, where z≧1 is an arbitrary parallelization factor that is chosen. The number of additional qubits to compute the (R-μ) term and sum in Equation (33) is determined by the number of additional qubits for each R i Note that these are not included because they can be done in place using existing registers to hold R. This is done by using R after computing the sum and exponential functions in equation (11) (which could be done, for example, before computing the sum). i This is possible because the value of is not used again.

[0662] Then the total T depth of the Riemann summation path loading process of precision ε for d assets and T time steps using registers of size (n,p) is

[0663] T RS (n,p,d,T,ε)=n 2 +2n 2 d2 / d+10(d2+d)+10T+9n+5+1.15nlog2(n / ε)+2T exp (n,p,ε)+T arcsin (n,p,ε), (85) is.

[0664] Here, T for ε exp and T arcsin The dependence of indicates that the polynomial approximation parameters k and M in equation (69) for each function depend on the target precision of the process. The total number of qubits involved is

[0665] q RS (n,p,d,T,ε)=Tn(4d+d2)+3n+1+q exp (n,p,ε)(1+dT)+q arcsin (n,p,ε). (86) is.

[0666] 9.2 Importance sampling for normalization in Riemann summation

[0667] In this section, we introduce a technique closely related to classical importance sampling to overcome the exponential scaling problem presented in Algorithm 3.1. The main idea is to approximate the target distribution by another distribution that can be loaded efficiently, and use quantum arithmetic only to adjust the (e.g., multiplicative) error.

[0668] P>1, and

[0669]

number

[0670] Assume a univariate probability density function f : [0,1] → [0,P] where f(x) is the probability density function of the n+2 qubits, and a payoff function g : [0,1] → [0,1]. In the considered context, g is applied only once. Therefore, it can be assumed to take values ​​in [0,1] without changing the overall complexity of the approach. As introduced earlier, consider a scaled function f(x) / P and corresponding operator F, as well as a corresponding operator G, to prepare a state with n+2 qubits given by:

[0671]

number

[0672] where x i = i / N. Then the probability of measuring |11〉 of the last two qubits is

[0673]

number

[0674] and when multiplied by P, if X~f,

[0675]

number

[0676] corresponds to a Riemann sum that approximates

[0677] Furthermore, the probability distribution h(x) can be efficiently loaded into the quantum state. i )∈[0,1]. That is, there is a known way to efficiently construct the quantum operator H as follows:

[0678]

number

[0679] Assume that there exists h such that for all x, f(x) / (h(x)N)∈[0,1], and then define a new operator F h Configure:

[0680]

number

[0681] H and F h Combining these gives us:

[0682]

number

[0683] This means that the probability of measuring |11〉 of the last two qubits is given by:

[0684]

number

[0685] That is, when X~f

[0686]

number

[0687] Therefore, if we obtain such a probability distribution h, we can multiply it by P and then, without rescaling,

[0688]

number

[0689] It can be easily seen that if P≦1, one can set h(x)=1 / N and recover the original approach without importance sampling.

[0690] For multivariate probability density functions, three cases are distinguished: First, for t = 0, ..., T, we have a univariate function f t In this case, the univariate approach can be applied directly, and each f t The corresponding h t Second, P>1 and

[0691]

number

[0692] A non-separable multivariate probability density function f:[0,1] where d →[0,P]. Each dimension is discretized using n qubits, i.e., N d Then, for all x, f(x) / (h(x)N d )∈[0,1], and the analysis is similar to the univariate case. Finally, consider the case of a multivariate probability density function obtained from a stochastic process and given by:

[0693]

number

[0694] where x t ∈[0,1] d , and f0(x0), f for t=0,…,T t (x t |x t-1 )∈[0,P]. Assume the following separable probability distributions:

[0695]

number

[0696] This can be efficiently loaded and the corresponding decomposition is

[0697]

number

[0698] where:

[0699]

number

[0700] At this time,

[0701]

number

[0702] is.

[0703] Therefore, h t If we find an h such that can be loaded efficiently,

[0704]

number

[0705]

number

[0706] and then the exponential scaling overhead P T+1 Efficiently loads stochastic processes without P. Again, for P ≤ 1,

[0707]

number

[0708] is set,

[0709]

number

[0710] and it can be easily seen that the original approach can be recovered without importance sampling.

[0711] Note that even if it is not possible to find an h that satisfies all criteria, the approach just described can still help reduce the overhead resulting from scaling.

[0712] 10.0 Reparameterized Pathway Loading Resource Estimation

[0713] To prepare a standard normal distribution that can be used by the quantum resource estimation system 102 in implementing the reparameterized loading approach of the subject disclosure, the quantum resource estimation system 102 (e.g., via the variational component 202) can utilize the variational methods described in Section 3.2.1 and corresponding gate and / or qubit costs, depending on the desired accuracy of the approximation. In addition, as described in Algorithm 3.2, affine transformations

[0714]

number

[0715] Note that the affine transformation is used to calculate asset prices from log returns, which for asset j at time t' is:

[0716]

number

[0717] where:

[0718]

number

[0719] is sum

[0720]

number

[0721] is the i-th component of (98). One complication in equation (98) is that the asset prices cannot be calculated completely in parallel among the l assets because the logarithmic returns of any correlated assets contribute to the calculation of each other's price. If all assets are pairwise correlated, the quantum resource estimation system 102 can calculate the contribution to the price of each asset from the logarithmic returns of all l assets at that time step, and calculate all asset prices for each time step by using the sum d 2 However, the quantum resource estimation system 102 can perform d additions in parallel, where the contribution of asset j's return to the price of asset (j+i)%d is calculated for any choice of i∈[0,d−1], since all d such operations have distinct source and target registers. Then, by d additions, the term

[0722]

number

[0723] is calculated and the quantum resource estimation system 102 calculates in a separate register for each t'

[0724]

number

[0725] When computing ∑ i = 1 ⁢ ...

[0726] The arithmetic error in computing Equation (98) can be reduced (e.g., minimized) by increasing the qubit register size to the largest possible value for the sum over time step T and asset d. If each Gaussian prepared in Equation (38) is discretized using n qubits, then:

[0727]

number

[0728] qubits are expressed as

[0729]

number

[0730] It is sufficient to hold the maximum value of

[0731]

number

[0732] The qubit has coefficients for all i,j.

[0733]

number

[0734] Assuming that,

[0735]

number

[0736] This condition is true for (e.g., the definition |ρ ij |≦1) covariance matrix Σ ij =Δtρ ij σ i σ jIt is not difficult to meet the typical situation of practical interest, which can be argued by looking at the elements of σ. Typically, the annualized volatility is less than 100% (i.e., σ i <1), the time step typically satisfies Δt<1, which means that the price of the underlying asset can be sampled more frequently than once a year. However, if neither condition is met, the quantum resource estimation system 102 may increase |Σ ij A smaller Δt can be chosen to ensure |<1.

[0737] In the reparameterization approach for derivatives defined over d assets and T time steps, the contributions to the T-depth and number of qubits for loading paths and computing asset prices are:

[0738] to load the Gaussian state in Eq. (38) using the variational method of Section 3.2.1.

[0739]

number

[0740] Depth T, where each Gaussian is prepared in parallel and the variational Ansatz has depth L. This step involves T*d*n qubits, where n qubits are used to prepare each Gaussian state.

[0741] Additional

[0742]

number

[0743] In equation (98), which contains qubits, for t'∈[2,T], all the sums

[0744]

number

[0745] T to calculate add *(T-1).

[0746] In equation (98),

[0747]

number

[0748] and

[0749]

number

[0750] T to calculate all the contributions for the additional qubits add *d.

[0751] in equation (98) over assets and time steps

[0752]

number

[0753] T for calculating contribution add .

[0754] T for calculating the exponential function in equation (98) over assets and time steps exp , and q given by Eq. (70) exp With q exp *d*T additional qubits.

[0755] Overall, the total T depth for path loading using the reparameterization method with precision ε for d assets and T time steps is

[0756] T RP(n,d,T,L,ε)=1.15nlog2(n / ε)(L+1)+10(d+T)+T exp (n,ε), (99)

[0757] where the number of qubits is

[0758]

number

[0759] where:

[0760]

number

[0761] is.

[0762] 11.0 Methods for Gaussian Loader Training

[0763] This section describes an approximate method for initializing quantum registers using a variational quantum eigensolver (VQE) approach. The algorithm features a parameterized circuit that generates a parameterized state |ψ({θ})〉 that approximately represents a target state |φ0〉, and updates its parameters {θ} to optimize the expected value of a suitable cost function. We show that the choice of the optimal cost function is critical to successful training.

[0764] Energy-Based Training

[0765] First, the quantum resource estimation system 102 may employ a physics-based approach and define an operator H such that its expected E value assumes its lowest possible value E when evaluated at the target state, as follows:

[0766] E0=〈Ψ0|H|Ψ0〉 (101)

[0767] In physics, the operator H is usually called the Hamiltonian, E is the energy, and |φ0〉 is the ground state.

[0768]

number

[0769] It is well known that is the ground state of the quantum harmonic oscillator Hamiltonian:

[0770] H=P 2 2m+m(X-x0) 2 2, (103)

[0771] where X is the position operator in real space, P = -iddx is the momentum operator, m is a parameter that determines the variance of the desired Gaussian distribution, and x0 is the center of the Gaussian distribution.

[0772]

number

[0773] To find the state φ(x) such that 2 ) can be adjusted.

[0774] Note that it is always possible to find the generating Hamiltonian function such that the ground state is the square root of the smooth distribution function to be loaded (eg, the target probability distribution).

[0775] To translate such considerations into an operational workflow, the quantum resource estimation system 102 can define how to compute the expectation value of equation (103) using a quantum computer. To this end, the operator X 2can be observed to be diagonal in the computational basis, which means that the number of bit string histograms N generated by repeated wave function collapses count (j) can be measured directly from the operator P 2 is diagonal in the momentum basis. This implies the addition of a centered quantum Fourier transform (QFT) circuit after the state preparation block. The quantum resource estimation system 102 can use a centered Fourier transform to allow for negative momenta. As mentioned above, the quantum resource estimation system 102 calculates the quantum state in the discrete position space x i = -w+iΔx, where i=0,…2 n -1, and Δx=2w / 2 n Without loss of generality, the quantum resource estimation system 102 may select a region centered around zero.

[0776]

number

[0777] can be calculated as follows:

[0778]

number

[0779]

number

[0780] where N shots is the total number of circuit iterations in the space and momentum basis. counts (j) (where 0≦N counts (j)≦N shots ,Σ j N counts (j)=N shots) is the number of measurements collapsed to the qubit basis state corresponding to the binary representation of the integer j. This strategy avoids the use of the Pauli representation of equation (103), which would involve an exponentially growing Pauli string scaled by the qubit register size.

[0781] The first step of the program is to numerically verify the possibility of preparing states that systematically converge to equation (102) using quantum circuits. By adopting a variational approach, we avoid expensive quantum arithmetic operations at the expense of introducing a source of error that is always present in numerical variational approaches. The most trivial one concerns the possibility of getting trapped in a local minimum during the (classical) optimization procedure. The second, and more serious, one is related to the expressiveness of the trial states generated by (e.g., shallow) quantum circuits.

[0782] The main choice for Ansatz is the so-called R y -CNOT circuit. The quantum resource estimation system 102

[0783]

number

[0784] The initial state defined on the n-qubit register can be set to unitary.

[0785]

number

[0786] is expanded under the action of

[0787]

number

[0788] Give.

[0789] This circuit uses an entangler U that spans the length of the qubit register. ENT followed by a single qubit rotation

[0790]

number

[0791] In the experimental case study described above, the quantum resource estimation system 102 estimates the quantum bit q i is the quantum bit q i-1 and the control qubit q i+1 Select a ladder of CNOT gates with linear connectivity so that it is the target of U (where i = 1, ..., n-2). Finally, U R One additional layer of gates is applied, bringing the number of variational parameters to n×(L+1).

[0792] Since single qubit rotations are all local operations,

[0793]

number

[0794] can be written as a tensor product of single-qubit rotations as follows:

[0795]

number

[0796]

number

[0797] is the quantum bit q i is a rotation about the Y axis on the Bloch sphere of , k=1,…,L+1. The complete unitary circuit operation is described by:

[0798]

number

[0799] And the parameterized state is:

[0800]

number

[0801] unitary

[0802]

number

[0803] Note that describes the complete circuit, but does not describe pre-measurement changes in the basis that can be included to collapse the wave function in momentum space as explained above.

[0804] For each n and L parameter, the quantum resource estimation system 102 may repeat the optimization run, e.g., eight times, to collect sufficient statistics because the optimization may get stuck in a suboptimal local minimum. Because the quantum resource estimation system 102 can use a classical emulation of the quantum circuit, the only source of error in the optimization comes from the classical optimizer. In experimental runs performed by the quantum resource estimation system 102 to implement the case study described above, the quantum resource estimation system 102 first performed a warm-up run with the COBYLA optimizer, followed by a longer run using the BFGS optimizer. To improve the efficiency of the optimization, the starting point for the VQE run at depth L uses the optimal parameters found in previous optimizations at depths L-2 or L-1, if available. Note that the portion of the algorithm related to classical optimization feedback can be significantly improved, for example, using gradient-based methods or imaginary-time-inspired update schemes.

[0805] L ∞ training refinement

[0806] As described above, the quantum resource estimation system 102 uses L as a cost function. ∞ The pre-optimized circuit obtained using the energy optimization method can be used as a starting guess for re-optimization according to

[0044] . In Figures 6A, 6B, and 6C, the direct L ∞ It is shown how optimization does not always provide acceptable results.

[0807] 6A, 6B, and 6C illustrate exemplary non-limiting graphs 600a, 600b, and 600, respectively, that can facilitate estimation of quantum resources for calculating the expectation value of a stochastic process using a reparameterization method according to one or more embodiments described herein. Repetitive descriptions of similar elements and / or processes utilized in each embodiment have been omitted for the sake of brevity.

[0808] In each of the graphs 600a, 600b, and 600c shown in Figures 6A, 6B, and 6C, respectively, the optimization run obtained using the energy-based method is represented by plot 602. In each of the graphs 600a, 600b, and 600c shown in Figures 6A, 6B, and 6C, respectively, the direct L ∞ The optimization is represented by plot 604. In each of the graphs 600a, 600b, and 600c shown in Figures 6A, 6B, and 6C, respectively, L ∞ A mixed strategy in which the energy-based optimization is further refined using optimization is represented by plot 606. To obtain the data plotted in each of graphs 600a, 600b, and 600c, quantum resource estimation system 102 may perform eight independent runs with the given parameters n qubits and L.

[0809] The complete results of the optimization are shown in Figures 7A and 7B.

[0810] 7A and 7B illustrate exemplary non-limiting graphs 700a and 700b, respectively, that can facilitate estimation of quantum resources for calculating the expectation value of a stochastic process using a reparameterization method according to one or more embodiments described herein. Repetitive descriptions of similar elements and / or processes utilized in each embodiment have been omitted for the sake of brevity.

[0811] Graph 700a shows the L between the prepared distribution and the target distribution. ∞ The norm difference is shown as a function of circuit depth L for different qubit register sizes n. The best of eight independent optimizations for each parameter is plotted in graph 700a. Plot 702 in graph 700a corresponds to an optimization performed using the energy of the quantum harmonic oscillator as the cost function. Plot 704 in graph 700a corresponds to an optimization performed using L as the cost function. ∞ Supports sophisticated optimizations using

[0812] Graph 700b shows the difference in energy of the associated quantum harmonic oscillator model as a function of circuit depth L for different qubit register sizes n. The best of eight independent optimizations for each parameter is plotted in graph 700b. Plot 702 in graph 700b corresponds to an optimization performed using the quantum harmonic oscillator energy as a cost function. Plot 704 in graph 700b corresponds to an optimization performed using L as a cost function. ∞ As expected, L ∞ Refinement targeting ∑ ...

[0813] L ∞ Norm Direct optimization failure

[0814] Here, an empirical explanation for the observed failure of direct norm optimization techniques is provided. To this end, the quantum resource estimation system 102 combines energy-based methods with direct L ∞ The quantum resource estimation system 102 can explore the cost function landscape for both optimization and optimization methods.

[0815]

number

[0816] Starting from , we can make cuts in the parameter space according to the following prescription:

[0817]

number

[0818] where:

[0819]

number

[0820] is a vector containing uniformly distributed random numbers in the range [-1, 1], and λ∈[-π,π] is a scalar that parameterizes the deformation from the optimal solution.

[0821] 8A and 8B illustrate exemplary non-limiting graphs 800a and 800b, respectively, that can facilitate estimation of quantum resources for calculating the expectation value of a stochastic process using a reparameterization method according to one or more embodiments described herein. Repetitive descriptions of similar elements and / or processes utilized in each embodiment have been omitted for the sake of brevity.

[0822] Graphs 800a and 800b shown in FIGS. 8A and 8B, respectively, show three different cut directions (e.g., vector

[0823]

number

[0824] 8 shows probing the cost function landscape for three different realizations of the energy E. The plots shown at the top of graphs 800a and 800b represent the cost function landscape versus energy E, while the plots shown at the bottom of graphs 800a and 800b represent the cost function landscape versus energy E calculated for two different settings n=5, 7 and depth L=6, 10, respectively, using three different cuts along the parameter space. ∞ Represents the cost function landscape for the norm.

[0825] From graphs 800a and 800b, L ∞ We observe that the cost function defined by the norm is much more corrugated than that defined by the energy E of the related quantum mechanical toy problem, which instead exhibits a smooth surface. ∞ The basins of attraction of the cost function overlap, because they are close to Gaussian functions for which the ground state of the physical problem is achieved, and therefore L ∞ The second optimization with the norm will not get stuck in high-cost local minima outside such basins.

[0826] Variational parameter digitization

[0827] The numerical results presented above provide evidence of a rather efficient Gaussian state preparation in terms of circuit depth for parameterized circuits, but additional steps need to be taken in terms of fault-tolerant implementation of such circuits. In this new framework, the continuous rotation RY gate can be extended as a finite product of discrete operations. Again following the Solov'y-Kitaev theorem or more specialized results, it is also possible to obtain an efficient representation of any SU(2) operator as a sequence of Clifford+T gates that scales logarithmically with the threshold error ε. In this regime where the angle of rotation can only take discretized values, it is investigated how the previously obtained results can be propagated. Thus, for each parameter

[0828]

number

[0829] is in the format i*2π / M digit It can be assumed that the vectors can only be expressed by ∑ i = ...

[0830] The quantum resource estimation system 102 can employ a protocol to optimize parameters on a lattice. First, the quantum resource estimation system 102 can project the original continuous-valued parameters onto a lattice that takes the closest lattice point for each parameter. Then, the quantum resource estimation system 102 compares the L ∞ A local search can be performed on the lattice to find a better combination of digitized parameters that reduces (e.g., minimizes) the norm difference. The quantum resource estimation system 102 can numerically show that the error introduced by such digitization systematically decreases with mesh size. The quantum resource estimation system 102 considers the L∞ norm difference error introduced by such digitization and shows that it is 1 / M digitIn all cases, the quantum resource estimation system 102 can obtain values ​​that are compatible with a continuous-valued solution, or the mesh size is M digit ~10 5 When it reaches , this becomes 2π / M digit A better value can be obtained, equivalent to discretizing the space at ≈0.0001 radians (rad).

[0831] 9A, 9B, and 9C illustrate exemplary non-limiting graphs 900a, 900b, and 900c, respectively, that can facilitate estimation of quantum resources for calculating the expectation value of a stochastic process using a reparameterization method according to one or more embodiments described herein. Repetitive descriptions of similar elements and / or processes utilized in each embodiment have been omitted for the sake of brevity.

[0832] Graph 900a shown in FIG. 9A shows the L between the prepared distribution and the target distribution. ∞ The norm difference is calculated for two different circuit depths L with n = 4 qubits and a digitized mesh size M digit As a function of each M digit For L, the quantum resource estimation system 102 can "digitize" the eight parameter sets obtained by a previous independent optimization (e.g., performed by considering a continuous range of rotation angle values). Empty (e.g., blank) square and circle symbols represent with respect to the entire data set, while non-empty symbols represent the minimum value in the set. Horizontal line 902 indicates the best value obtained in the previous optimization performed by considering a continuous range for rotation angle values ​​for each L parameter. In some instances, digitization helps to escape local minima and achieve a slightly better solution. Diagonal line 904 is a visual guide, and the function 1 / M digit and 0.1 / M digit Represents.

[0833] Graph 900b shown in FIG. 9B includes an example, non-limiting, alternative embodiment of graph 900a, where graph 900b shows the L between the prepared distribution and the target distribution.∞ The norm difference is calculated for two different circuit depths L with n = 5 qubits and a digitized mesh size M digit Graph 900c shown in FIG. 9C includes an example, non-limiting, alternative embodiment of graph 900a, where graph 900c illustrates the relationship between the prepared distribution and the target distribution. ∞ The norm difference is calculated for two different circuit depths L with n = 6 qubits and a digitized mesh size M digit is shown as a function of .

[0834] 10 illustrates a flow diagram of an exemplary, non-limiting, computer-implemented method 1000 that can facilitate estimation of quantum resources for calculating the expectation value of a stochastic process using a reparameterization method according to one or more aspects described herein. Repeated descriptions of similar elements and / or processes utilized in each aspect have been omitted for the sake of brevity.

[0835] At 1002, the computer-implemented method 1000 may include applying, by a system operably coupled to a processor (e.g., processor 106) (e.g., via quantum resource estimation system 102 and / or reparameterization component 108), a quantum fault-tolerant operation to a variationally prepared quantum state corresponding to a probability distribution to generate a quantum state corresponding to a target probability distribution.

[0836] At 1004, the computer-implemented method 1000 may include estimating, by the system (e.g., via the quantum resource estimation system 102 and / or the estimation component 110), at least one defined criterion for a quantum computer to be used to calculate an expected value of a stochastic process associated with a target probability distribution (e.g., the value of a derivative asset).

[0837] The quantum resource estimation system 102 may be associated with various technologies, for example, quantum computing technology, quantum hardware and / or software technology, quantum algorithm technology, machine learning technology, artificial intelligence technology, cloud computing technology, and / or other technologies.

[0838] The quantum resource estimation system 102 may provide technical improvements to systems, devices, components, operational steps, and / or processing steps associated with the various technologies identified above. For example, the quantum resource estimation system 102 may apply quantum fault-tolerant operations to variationally prepared quantum states corresponding to a probability distribution to generate quantum states corresponding to a target probability distribution and / or estimate at least one defined criterion of a quantum computer used to calculate the expected value of a stochastic process associated with the target probability distribution (e.g., the value of a derivative asset). In this example, the at least one defined criterion may include an attribute, condition, property, parameter, or configuration of the quantum computer that enables the quantum computer to achieve a defined quantum advantage in calculating the expected value of a stochastic process associated with the target probability distribution (e.g., the value of a derivative asset). In this example, the quantum resource estimation system 102 may thus be implemented to identify a quantum resource (e.g., a quantum computer, a quantum processor, and / or another quantum resource) that can leverage the benefits of quantum computing to calculate the expected value of a stochastic process (e.g., the value of a derivative asset such as an options contract), while incurring a minimal amount of computational cost compared to other quantum resources.

[0839] The quantum resource estimation system 102 can provide technical improvements to a processing unit (e.g., processor 106, a quantum processor, and / or another processor) associated with the quantum resource estimation system 102. For example, as described above, the quantum resource estimation system 102 can also estimate, and therefore determine, at least one defined criterion that can enable a quantum computer to exploit the benefits of quantum computing to calculate the expected value of a stochastic process (e.g., the value of a derivative asset such as an options contract), while incurring a minimal amount of computational cost compared to other quantum resources. In this example, a quantum processor within a quantum computer can be developed (e.g., planned, designed, and / or manufactured) and / or modified to include at least one defined criterion that can be estimated and / or identified by the quantum resource estimation system 102 to enable the quantum computer to calculate the expected value of a stochastic process (e.g., the value of a derivative asset such as an options contract), while incurring a minimal amount of computational cost.

[0840] A practical application of the quantum resource estimation system 102 is that it can be implemented using a classical computing device (e.g., a classical computer) to estimate at least one defined criterion that can enable a quantum computer to leverage the advantages of quantum computing to compute one or more solutions (e.g., heuristics) to a variety of complex problems (e.g., estimation problems, optimization problems, and / or other problems) in various domains (e.g., finance, chemistry, medicine, and / or other domains). For example, a practical application of the quantum resource estimation system 102 is that it can be implemented using a classical computing device (e.g., a classical computer) to estimate at least one defined criterion that can enable a quantum computer to leverage the advantages of quantum computing to compute one or more solutions (e.g., heuristics) to estimation and / or optimization problems in the domains of chemistry, medicine, and / or finance, where such solutions can be used, for example, to engineer new chemical compounds, new drugs, and / or new option premiums.

[0841] It should be appreciated that the quantum resource estimation system 102 provides a novel approach driven by relatively new quantum computing technology. For example, the quantum resource estimation system 102 provides a novel approach to estimating at least one defined criterion that can enable a quantum computer to leverage the benefits of quantum computing to calculate the expected value of a stochastic process (e.g., the value of a derivative asset such as an options contract), while incurring a minimal amount of computational cost compared to other quantum resources.

[0842] The quantum resource estimation system 102 may utilize hardware or software to solve problems that are highly technical in nature, not abstract, and cannot be performed as a set of mental acts by a human. In some aspects, one or more of the processes described herein may be executed by one or more specialized computers (e.g., specialized processing units, specialized classical computers, specialized quantum computers, and / or another type of specialized computer) to perform defined tasks associated with the various technologies identified above. The quantum resource estimation system 102 and / or its components may be utilized to solve new problems that arise through the utilization of the above-described technological advances, quantum computing systems, cloud computing systems, computer architectures, and / or other technologies.

[0843] It should be understood that the quantum resource estimation system 102 may utilize various combinations of electrical components, mechanical components, and circuitry that cannot be replicated by or performed by a human mind, as various operations capable of being performed by the quantum resource estimation system 102 and / or its components described herein are operations that are beyond the capabilities of human intelligence. For example, the amount of data processed by the quantum resource estimation system 102 in a particular period of time, the rate at which such data is processed, or the types of data may be greater in amount, greater in speed, or different in type from what a human mind could process in the same period of time.

[0844] According to some aspects, the quantum resource estimation system 102 may also be fully operable to perform one or more other functions (e.g., fully powered on, fully running, and / or another function) while performing various operations described herein. It should be understood that performing such simultaneous operations exceeds human intellectual capabilities. It should also be understood that the quantum resource estimation system 102 may include information that an entity, such as a human user, cannot manually obtain. For example, the type, amount, and / or variety of information included in the quantum resource estimation system 102, the reparameterization component 108, the estimation component 110, the variation component 202, and / or the error analysis component 302 may be more complex than information manually obtained by a human user.

[0845] In some aspects, the quantum resource estimation system 102 may be associated with a cloud computing environment. For example, the quantum resource estimation system 102 may be associated with the cloud computing environment 1250 described below with reference to FIG. 12 and / or one or more functional abstraction layers (e.g., hardware and software layer 1360, virtualization layer 1370, management layer 1380, and / or workload layer 1390) described below with reference to FIG.

[0846] Quantum resource estimation system 102 and / or its components (e.g., reparameterization component 108, estimation component 110, variational component 202, error analysis component 302, and / or another component) may utilize one or more computing resources of a cloud computing environment 1250, described below with reference to FIG. 12 , and / or one or more functional abstraction layers (e.g., quantum software), described below with reference to FIG. 13 , to perform one or more operations in accordance with one or more aspects of the subject disclosure described herein. For example, cloud computing environment 1250 and / or such one or more functional abstraction layers may include one or more classical computing devices (e.g., a classical computer, a classical processor, a virtual machine, a server, and / or another classical computing device), quantum hardware and / or quantum software (e.g., a quantum computing device, a quantum computer, a quantum processor, a quantum circuit simulation software, a superconducting circuit, and / or other quantum hardware and / or quantum software), which may be utilized by quantum resource estimation system 102 and / or its components to perform one or more operations in accordance with one or more aspects of the subject disclosure described herein. For example, the quantum resource estimation system 102 and / or its components may utilize such one or more classical and / or quantum computing resources to perform one or more classical and / or quantum: mathematical functions, calculations, and / or equations; computing and / or processing scripts; algorithms; models (e.g., artificial intelligence (AI) models, machine learning (ML) models, and / or other types of models); and / or other operations in accordance with one or more aspects of the subject disclosure described herein.

[0847] While the subject disclosure includes detailed descriptions of cloud computing, it should be understood that implementation of the teachings described herein is not limited to cloud computing environments. Rather, aspects of the present invention can be implemented in connection with any other type of computing environment now known or later developed.

[0848] Cloud computing is a model of service delivery for enabling convenient, on-demand network access to a shared pool of configurable computing resources (e.g., networks, network bandwidth, servers, processing, memory, storage, applications, virtual machines, and services) that can be rapidly provisioned and released with minimal management effort or interaction with the provider of the service. The cloud model just described may include at least five characteristics, at least three service models, and at least four deployment models.

[0849] Its features are as follows:

[0850] On-demand self-service: Cloud consumers can automatically and unilaterally provision computing capacity, such as server time or network storage, as needed, without the need for human interaction with the service provider.

[0851] Broad Network Access: Functionality is available over the network and accessed through standard mechanisms that facilitate use by heterogeneous thin or thick client platforms (eg, mobile phones, laptops, and PDAs).

[0852] Resource Pooling: To serve a large number of consumers using a multi-tenant model, a provider's computing resources are pooled, with different physical and virtual resources dynamically allocated and reallocated on demand. Consumers generally have no control over or knowledge of the exact location of the resources provided, but there is a sense of location independence in the sense that they may be able to specify the location at a higher level of abstraction (e.g., country, state, or data center).

[0853] Rapid Elasticity: Capabilities can be quickly and elastically provisioned, quickly scaled out in some cases automatically, and quickly released and quickly scaled in. To the consumer, the capabilities available for provisioning often appear unlimited and can be purchased at any time and in any quantity.

[0854] Metered Services: Cloud systems leverage metering capabilities to automatically control and optimize resource usage at several levels of abstraction appropriate to the type of service (e.g., storage, processing, bandwidth, and active user accounts). Resource usage can be monitored, controlled, and reported, providing transparency to both providers and consumers of the services used.

[0855] The service model is as follows:

[0856] Software as a Service (SaaS): The functionality offered to the consumer is the use of the provider's applications running on a cloud infrastructure. The applications are accessible from a variety of client devices through thin-client interfaces such as web browsers (e.g., web-based email). The consumer does not manage or control the underlying cloud infrastructure, including the network, servers, operating systems, storage, or individual application functions, with the possible exception of limited user-specific application configuration settings.

[0857] Platform as a Service (PaaS): The capability offered to consumers is to deploy consumer-created or acquired applications, written using programming languages ​​and tools supported by the provider, onto a cloud infrastructure. The consumer does not manage or control the underlying cloud infrastructure, including the network, servers, operating systems, or storage, but does control the deployed applications and, in some cases, the application-hosting environment configuration.

[0858] Infrastructure as a Service (IaaS): The functionality provided to consumers is the provisioning of processing, storage, network, and other basic computing resources, allowing the consumer to deploy and run any software, which may include operating systems and applications. The consumer does not manage or control the underlying cloud infrastructure, but rather controls the operating systems, storage, and deployed applications, and in some cases has limited control over selected networking components (e.g., host firewalls).

[0859] The deployment models are as follows:

[0860] Private Cloud: Cloud infrastructure is operated solely for the organization. It may be managed by the organization or a third party and may reside on-premises or off-premises.

[0861] Community Cloud: Cloud infrastructure is shared among multiple organizations to support a specific community with shared interests (mission, security requirements, policies, compliance considerations). It may be managed by the organization or a third party and may reside on-premises or off-premises.

[0862] Public cloud: Cloud infrastructure is made available to the public or large industry groups and is owned by organizations that sell cloud services.

[0863] Hybrid Cloud: A cloud infrastructure is a composition of two or more clouds (private, community, or public) that are joined together while remaining a unique entity by standardized or proprietary technologies that allow for the portability of data and applications (e.g., cloud bursting for load balancing between clouds).

[0864] Cloud computing environments are service-oriented with an emphasis on statelessness, low coupling, modularity, and semantic interoperability. At the heart of cloud computing is an infrastructure that comprises a network of interconnected nodes.

[0865] For ease of explanation, computer-implemented methodologies are depicted and described as a series of acts. It is to be understood and appreciated that the subject innovation is not limited by the depicted acts and / or the order of acts; for example, acts can occur in various orders and / or simultaneously, and with other acts not presented or described herein. Moreover, not all depicted acts are necessarily required to implement a computer-implemented methodology in accordance with the disclosed subject matter. In addition, those skilled in the art will understand and appreciate that a computer-implemented methodology can alternatively be represented as a series of interrelated states via a state diagram or events. Furthermore, it should be further appreciated that the computer-implemented methodologies disclosed hereinafter and throughout this specification can be stored on an article of manufacture to facilitate transportation and transfer of such computer-implemented methodologies to a computer. The term article of manufacture, as used herein, is intended to encompass a computer program accessible from any computer-readable device or storage medium.

[0866] To provide a context for various aspects of the disclosed subject matter, Figure 11 and the following discussion are intended to provide a general description of a suitable environment in which various aspects of the disclosed subject matter may be implemented. Figure 11 illustrates a block diagram of an exemplary, non-limiting operating environment in which one or more aspects described herein may be facilitated. Repeated descriptions of similar elements utilized in other aspects described herein have been omitted for the sake of brevity.

[0867] 11 , a suitable operating environment 1100 for implementing various aspects of the subject disclosure may also include a computer 1112. The computer 1112 may also include a processing unit 1114, a system memory 1116, and a system bus 1118. The system bus 1118 couples system components including, but not limited to, the system memory 1116 to the processing unit 1114. The processing unit 1114 may be any of a variety of available processors. Dual microprocessors and other multi-processor architectures may also be utilized as the processing unit 1114. The system bus 1118 may be any of several types of bus structures, including a memory bus or memory controller, a peripheral bus or external bus, and / or a local bus using any of a variety of available bus architectures, including, but not limited to, Industry Standard Architecture (ISA), Micro Channel Architecture (MSA), Enhanced ISA (EISA), Intelligent Drive Electronics (IDE), VESA Local Bus (VLB), Peripheral Component Interconnect (PCI), Card Bus, Universal Serial Bus (USB), Advanced Graphics Port (AGP), Firewire (IEEE 1394), and Small Computer System Interface (SCSI).

[0868] The system memory 1116 may also include volatile memory 1120 and nonvolatile memory 1122. A basic input / output system (BIOS), containing the basic routines for transferring information between elements within the computer 1112, such as during start-up, is stored in the nonvolatile memory 1122. The computer 1112 may also include removable and non-removable, volatile and non-volatile computer storage media. FIG. 11 illustrates, for example, disk storage 1124. The disk storage 1124 may include devices such as, but not limited to, magnetic disk drives, floppy disk drives, tape drives, Jaz drives, Zip drives, LS-100 drives, flash memory cards, memory sticks, and the like. The disk storage 1124 may also include storage media, either separately or in combination with other storage media. A removable or non-removable interface, such as interface 1126, is typically used to facilitate connection of the disk storage 1124 to the system bus 1118. 11 also illustrates software that acts as an intermediary between users and the basic computer resources described in suitable operating environment 1100. Such software may include, for example, operating system 1128. Operating system 1128, which may be stored on disk storage 1124, acts to control and allocate resources of the computer 1112.

[0869] System applications 1130 take advantage of the management of resources by operating system 1128 through program modules 1132 and program data 1134, stored, for example, in either system memory 1116 or disk storage 1124. It is understood that the subject disclosure can be implemented with various operating systems or combinations of operating systems. Users enter commands or information into computer 1112 through input devices 1136. Input devices 1136 include, but are not limited to, pointing devices such as a mouse, trackball, stylus, touchpad, keyboard, microphone, joystick, gamepad, satellite dish, scanner, TV tuner card, digital camera, digital video camera, webcam, and the like. These and other input devices are connected to processing unit 1114 through system bus 1118 via interface ports 1138. Interface ports 1138 include, for example, serial ports, parallel ports, game ports, and universal serial bus (USB). Output devices 1140 use some of the same types of ports as input devices 1136. Thus, for example, a USB port may be used to provide input to computer 1112, and to output information from computer 1112 to output device 1140. Output adapter 1142 is provided to illustrate that there are some output devices 1140 such as monitors, speakers, printers, among other output devices 1140, that may require special adapters. Output adapters 1142 include, by way of example and not limitation, video cards and sound cards, which provide a means of connection between output device 1140 and system bus 1118. It should be noted that other devices and / or systems of devices provide both input and output capabilities, such as remote computer(s) 1144.

[0870] The computer 1112 can operate in a networked environment using logical connections to one or more remote computers, such as a remote computer 1144. The remote computer 1144 can be a computer, a server, a router, a network PC, a workstation, a microprocessor-based device, a peer device or other common network node, and typically includes many or all of the elements described relative to the computer 1112. For simplicity, only a memory storage device 1146 is shown with the remote computer 1144. The remote computer 1144 is logically connected to the computer 1112 through a network interface 1148 and physically connected via a communication connection 1150. The network interface 1148 encompasses wired and / or wireless communication networks, such as a local area network (LAN), a wide area network (WAN), a cellular network, and / or another wired and / or wireless communication network. LAN technologies include Fiber Distributed Data Interface (FDDI), Copper Distributed Data Interface (CDDI), Ethernet, and Token Ring. WAN technologies include, but are not limited to, point-to-point links, integrated services digital networks (ISDN) and variations thereon, packet-switched networks, and circuit-switched networks such as digital subscriber lines (DSL). Communications connection(s) 1150 refer to the hardware / software utilized to connect network interface 1148 to system bus 1118. While communications connection(s) 1150 are shown internal to computer 1112 for clarity, they may also be external to computer 1112. The hardware / software for connecting to network interface 1148 may also include, by way of example only, internal and external technologies such as modems, including ordinary telephone-grade modems, cable modems, and DSL modems, ISDN adapters, and Ethernet cards.

[0871] Referring now to FIG. 12 , an exemplary cloud computing environment 1250 is shown. As shown, the cloud computing environment 1250 includes one or more cloud computing nodes 1210 with which local computing devices used by cloud consumers, such as, for example, a personal digital assistant (PDA) or mobile phone 1254A, a desktop computer 1254B, a laptop computer 1254C, and / or an automobile computer system 1254N, may communicate. Although not shown in FIG. 12 , the cloud computing node 1210 may further include a quantum platform (e.g., a quantum computer, quantum hardware, quantum software, and / or another quantum platform) with which the local computing devices used by the cloud consumers may communicate. The nodes 1210 may communicate with each other. They may be grouped (not shown) in one or more networks, such as the private, community, public, or hybrid clouds described herein above, or combinations thereof, either physically or virtually. This allows the cloud computing environment 1250 to provide infrastructure, platform, and / or software as a service without the need for cloud consumers to maintain resources on their local computing devices. It will be understood that the types of computing devices 1254A-N shown in FIG. 12 are intended to be illustrative only, and that the computing node 1210 and cloud computing environment 1250 can communicate with any type of computerized device via any type of network and / or network-addressable connection (e.g., using a web browser).

[0872] Referring now to Figure 13, a set of functional abstraction layers provided by cloud computing environment 1250 (Figure 12) is shown. It should be understood in advance that the components, layers, and functions shown in Figure 13 are intended to be illustrative only, and aspects of the present invention are not limited thereto. As shown, the following layers and corresponding functions are provided:

[0873] Hardware and software layer 1360 includes hardware and software components. Examples of hardware components include mainframe 1361, RISC (reduced instruction set computer) architecture-based server 1362, server 1363, blade server 1364, storage device 1365, and network and network components 1366. In some aspects, software components include network application server software 1367, database software 1368, quantum platform routing software (not shown in FIG. 13), and / or quantum software (not shown in FIG. 13).

[0874] The virtualization layer 1370 provides an abstraction layer in which the following examples of virtual entities may be provided: virtual servers 1371, virtual storage 1372, virtual networks including virtual private networks 1373, virtual applications and operating systems 1374, and virtual clients 1375.

[0875] In one example, management layer 1380 may provide the functions described below. Resource provisioning 1381 provides dynamic procurement of computing and other resources utilized to execute tasks within the cloud computing environment. Metering and pricing 1382 provides cost tracking as resources are utilized within the cloud computing environment and billing or invoicing for the consumption of those resources. In one example, these resources may include application software licenses. Security provides identity verification of cloud consumers and tasks, as well as protection of data and other resources. User portal 1383 provides consumers and system administrators with access to the cloud computing environment. Service level management 1384 provides allocation and management of cloud computing resources so that requested service levels are met. Service level agreement (SLA) planning and fulfillment 1385 provides proactive coordination and procurement of cloud computing resources in anticipation of future demand in accordance with SLAs.

[0876] The workload layer 1390 provides examples of functionality for which a cloud computing environment may be utilized. Non-limiting examples of workloads and functions that may be provided from this layer include mapping and navigation 1391, software development and lifecycle management 1392, virtual classroom instructional delivery 1393, data analytics processing 1394, transaction processing 1395, and quantum resource estimation software 1396.

[0877] The present invention may be a system, a method, an apparatus, and / or a computer program product, at any possible level of technical detail. A computer program product may include a computer-readable storage medium (or media) having computer-readable program instructions for causing a processor to execute aspects of the present invention. A computer-readable storage medium may be a tangible device capable of holding and storing instructions for use by an instruction execution device. A computer-readable storage medium may be, for example, but is not limited to, an electronic storage device, a magnetic storage device, an optical storage device, an electromagnetic storage device, a semiconductor storage device, or any suitable combination of the above. A non-exhaustive list of more specific examples of computer-readable storage media may include portable computer diskettes, hard disks, random access memory (RAM), read-only memory (ROM), erasable programmable read-only memory (EPROM or flash memory), static random access memory (SRAM), portable compact disk read-only memory (CD-ROM), digital versatile disk (DVD), memory stick, floppy disk, punch cards, or mechanically encoded devices such as raised structures in grooves with instructions recorded thereon, and any suitable combination of the above. As used herein, computer-readable storage media should not be construed as being ephemeral signals per se, such as radio waves or other freely propagating electromagnetic waves, electromagnetic waves propagating through a waveguide or other transmission medium (e.g., light pulses passing through a fiber optic cable), or electrical signals transmitted over wires.

[0878] The computer-readable program instructions described herein can be downloaded to each computing / processing device from a computer-readable storage medium or to an external computer or external storage device via a network, such as the Internet, a local area network, a wide area network, and / or a wireless network. The network may include copper transmission cables, optical fiber transmissions, wireless transmissions, routers, firewalls, switches, gateway computers, and / or edge servers. A network adapter card or network interface of each computing / processing device receives the computer-readable program instructions from the network and forwards the computer-readable program instructions for storage in a computer-readable storage medium within the respective computing / processing device. The computer-readable program instructions for carrying out the operations of the present invention may be assembler instructions, instruction set architecture (ISA) instructions, machine instructions, machine-dependent instructions, microcode, firmware instructions, state setting data, configuration data for integrated circuits, or source code or object code written in any combination of one or more programming languages, including object-oriented programming languages ​​such as Smalltalk or C++, and procedural programming languages ​​such as the "C" programming language or similar programming languages. The computer-readable program instructions may execute entirely on the user's computer as a stand-alone software package, may execute partially on the user's computer, may execute partially on the user's computer and partially on a remote computer, or may execute entirely on a remote computer or server. In the latter scenario, the remote computer may be connected to the user's computer via any type of network, including a local area network (LAN) or a wide area network (WAN), or may be connected to an external computer (e.g., via the Internet using an Internet Service Provider).In some embodiments, electronic circuitry including, for example, a programmable logic circuit, a field programmable gate array (FPGA), or a programmable logic array (PLA) can execute computer readable program instructions by utilizing state information of the computer readable program instructions to personalize the electronic circuitry to perform aspects of the present invention.

[0879] Aspects of the present invention are described herein with reference to flowchart illustrations and / or block diagrams of methods, apparatus (systems), and computer program products according to embodiments of the present invention. It will be understood that each block of the flowchart illustrations and / or block diagrams, and combinations of blocks in the flowchart illustrations and / or block diagrams, can be implemented by computer-readable program instructions. These computer-readable program instructions can be provided to a processor of a general-purpose computer, a special-purpose computer, or other programmable data processing apparatus generating machine, such that the instructions, executed by the processor of the computer or other programmable data processing apparatus, create means for implementing the function(s) / act(s) specified in one or more blocks of the flowchart illustrations and / or block diagrams. These computer-readable program instructions, which can instruct a computer, programmable data processing apparatus, and / or other device to function in a particular manner, can also be stored on a computer-readable storage medium, such that the computer-readable storage medium having the instructions stored thereon comprises an article of manufacture containing instructions that implement aspects of the function(s) / act(s) specified in one or more blocks of the flowchart illustrations and / or block diagrams. Also, the computer-readable program instructions can be loaded into a computer, other programmable data processing apparatus, or other device to cause a series of operational acts to be performed on the computer, other programmable apparatus, or other device to generate a computer-implemented process, such that the instructions executing on the computer, other programmable apparatus, or other device implement the functions / acts specified in one or more blocks of the flowcharts and / or block diagrams.

[0880] The flowcharts and block diagrams in the figures illustrate the architecture, functionality, and operation of possible implementations of systems, methods, and computer program products according to various aspects of the present invention. In this regard, each block in a flowchart or block diagram may represent a module, segment, or portion of an instruction, which comprises one or more executable instructions for implementing the specified logical function(s). In some alternative implementations, the functions noted in the blocks may occur out of the order noted in the figures. For example, two blocks shown in succession may in fact be executed substantially concurrently, or the blocks may be executed in the reverse order, depending on the functionality involved. It should also be noted that each block of the block diagrams and / or flowchart diagrams, and combinations of blocks in the block diagrams and / or flowchart diagrams, may be implemented by a special-purpose hardware-based system that performs the specified functions or acts or executes a combination of special-purpose hardware and computer instructions.

[0881] While the subject matter has been described above in the general context of computer-executable instructions for a computer program product executed on one computer and / or multiple computers, those skilled in the art will recognize that the subject disclosure may also be implemented or practiced in combination with other program modules. Generally, program modules include routines, programs, components, data structures, and / or other program modules that perform particular tasks and / or implement particular abstract data types. Furthermore, those skilled in the art will appreciate that the computer-implemented methods of the present invention can be practiced using other computer system configurations, including single-processor or multiprocessor computer systems, minicomputer devices, mainframe computers, as well as computers, handheld computing devices (e.g., PDAs, phones), microprocessor-based or programmable consumer or industrial electronics, and the like. The illustrated aspects can also be practiced in distributed computing environments where tasks are performed by remote processing devices linked through a communications network. However, some, if not all, aspects of the subject disclosure can be practiced on stand-alone computers. In a distributed computing environment, program modules may be located in both local and remote memory storage devices. For example, in one or more aspects, computer-executable components may be executed from a memory that may include or consist of one or more distributed memory units. As used herein, the terms "memory" and "memory unit" are interchangeable. Furthermore, one or more aspects described herein may execute code of computer-executable components in a distributed manner, e.g., multiple processors coupled or cooperating to execute code from one or more distributed memory units. As used herein, the term "memory" may encompass a single memory or memory unit in one location or memories or memory units in one or more locations.

[0882] As used herein, terms such as “component,” “system,” “platform,” and “interface” can refer to and / or include computer-related entities or entities associated with an operating machine having one or more specific functionalities. The entities disclosed herein can be hardware, a combination of hardware and software, software, or software in execution. For example, a component can be, but is not limited to, a process running on a processor, a processor, an object, an executable, a thread of execution, a program, and / or a computer. By way of example, both an application running on a server and the server can be a component. One or more components can reside within a process and / or thread of execution, and components can be localized on one computer and / or distributed between two or more computers. In another example, each component can execute from various computer-readable media having various data structures stored thereon. Components can communicate via local and / or remote processes, such as via signals having one or more data packets (e.g., data from one component interacting with another component via signals having one or more data packets, in a local system, a distributed system, and / or across a network such as the Internet with other systems). As another example, a component may be a device having specific functionality provided by mechanical parts operated by electrical or electronic circuitry, which in turn is operated by a software or firmware application executed by a processor. In such cases, the processor may be internal or external to the device and may execute at least a portion of the software or firmware application.As yet another example, a component may be a device that provides certain functionality without mechanical parts through electronic components, and the electronic components may include a processor or other means for executing software or firmware that at least partially provides the functionality of the electronic component. In an aspect, a component may emulate an electronic component via, for example, a virtual machine in a cloud computing system.

[0883] Furthermore, the term "or" is intended to mean an inclusive "or" rather than an exclusive "or." That is, unless otherwise specified or clear from the context, "X uses A or B" is intended to mean any of the natural inclusive permutations. That is, if X uses A, if X uses B, or if X uses both A and B, then "X uses A or B" is satisfied in each of the foregoing cases. Furthermore, as used in this specification and the accompanying drawings, the articles "a" and "an" should generally be construed to mean "one or more" unless otherwise specified or clear from the context to mean singular. As used herein, the terms "example" and / or "exemplary" are utilized to mean serving as an example, instance, or illustration. For the avoidance of doubt, the subject matter disclosed herein is not limited by such examples. Additionally, any aspect or design described herein as "example" and / or "exemplary" is not necessarily to be construed as preferred or advantageous over other aspects or designs, and is not intended to exclude equivalent exemplary structures and techniques known to those skilled in the art.

[0884] As used herein, the term "processor" can refer to virtually any computing processing unit or device, including, but not limited to, a single-core processor, a single processor with software multithreading execution capabilities, a multi-core processor, a multi-core processor with software multithreading execution capabilities, a multi-core processor with hardware multithreading technology, a parallel platform, and a parallel platform with distributed shared memory. Additionally, a processor can refer to an integrated circuit, an application-specific integrated circuit (ASIC), a digital signal processor (DSP), a field-programmable gate array (FPGA), a programmable logic controller (PLC), a complex programmable logic device (CPLD), discrete gate or transistor logic, discrete hardware components, or any combination thereof designed to perform the functions described herein. Furthermore, a processor can utilize nanoscale architectures, such as, but not limited to, molecular and quantum dot-based transistors, switches, and gates, to optimize space usage or improve performance of user equipment. A processor can also be implemented as a combination of arithmetic processing units. In the subject disclosure, terms such as "store," "storage," "data store," "data storage," "database," and substantially any other information storage component associated with the operation and functionality of a component are utilized to refer to a "memory component," an entity embodied in a "memory," or a component that includes a memory. It is understood that the memory and / or memory components described herein may be either volatile memory or non-volatile memory, or may include both volatile and non-volatile memory.By way of example, and not limitation, non-volatile memory may include read-only memory (ROM), programmable ROM (PROM), electrically programmable ROM (EPROM), electrically erasable ROM (EEPROM), flash memory, or non-volatile random access memory (RAM) (e.g., ferroelectric RAM (FeRAM)). Volatile memory may include RAM, which may function as external cache memory, for example. By way of example, and not limitation, RAM is available in various forms, including synchronous RAM (SRAM), dynamic RAM (DRAM), synchronous DRAM (SDRAM), double data rate SDRAM (DDR SDRAM), enhanced SDRAM (ESDRAM), SyncLink DRAM (SLDRAM), direct Rambus RAM (DRRAM), direct Rambus dynamic RAM (DRDRAM), and Rambus dynamic RAM (RDRAM). Additionally, the disclosed memory components of systems or computer-implemented methods herein are intended to comprise, without being limited to, these and any other suitable types of memory.

[0885] What has been described above includes merely exemplary systems and computer-implemented methods. Of course, it is not possible to describe every conceivable combination of components or computer-implemented methods for purposes of describing the subject disclosure, but one of ordinary skill in the art will recognize that many more combinations and permutations of the subject disclosure are possible. Furthermore, to the extent that terms such as "including," "having," and "holding" are used in the detailed description, claims, appendices, and drawings, such terms are intended to be inclusive in the same manner as "comprising," as "comprising" is interpreted when employed as a transitional term in a claim.

[0886] The descriptions of various embodiments are presented for illustrative purposes and are not intended to be exhaustive or limiting of the disclosed embodiments. Many modifications and variations will be apparent to those skilled in the art without departing from the scope and spirit of the described embodiments. The terms used herein were selected to best explain the principles of the embodiments, practical applications or technical improvements to technology found in the market, or to enable those skilled in the art to understand the embodiments disclosed herein.

Claims

1. 1. A computer-implemented method comprising: applying, by a system operatively connected to a processor, a quantum operation to a first quantum state corresponding to a probability distribution to generate a second quantum state corresponding to a target probability distribution, said applying step comprising: Applying dT Gaussian operators G to ndT qubits of a quantum computer, [Equation 1] and configuring said first quantum state given by: [Equation 2] is the 2nd order of this multivariate standard Gaussian ndT the path of a discrete-time multivariate stochastic process running over n realizations, said Gaussian operator G loading a Gaussian distribution onto an n-qubit register; [Equation 3] denotes the corresponding probability, and n, d, and T are positive integers; adjusting each Gaussian distribution of the first quantum state by performing an affine transformation, such that the resulting second quantum state has a return path ω corresponding to the target probability distribution. R and probability p(ω R ) and Including steps The method comprises: Quantum register of said quantum computer [Equation 4] to [Equation 5] and calculating the Rotate the register value to the amplitude of the ancilla qubit [Equation 6] generating a second quantum state given by: performing amplitude estimation in the second quantum state to determine the probability that the ancilla qubit is |1〉; 20. The computer-implemented method of claim 19, further comprising:

2. 2. The computer-implemented method of claim 1, wherein performing the affine transformation generates the second quantum state corresponding to the target probability distribution having at least one of a defined mean of the target probability distribution, a defined standard deviation of the target probability distribution, or one or more explicit parameters that specify the target probability distribution.

3. 2. The computer-implemented method of claim 1, wherein the first quantum state is a superposition over possible paths of the discrete-time multivariate stochastic process.

4. 10. The computer-implemented method of claim 1, wherein the first quantum state is a variationally prepared quantum state prepared by running a trained variational quantum circuit.

5. training, by the system, the variational quantum circuit to prepare the first quantum state and to reduce the computational cost of quantum arithmetic operations performed by the quantum computer to calculate the expectation value of a stochastic process associated with the target probability distribution.

5. The computer-implemented method of claim 4, further comprising:

6. training, by the system, the variational quantum circuit to prepare the variationally prepared quantum state, the step including using a Hamiltonian operator to generate a basis state corresponding to the target probability distribution.

5. The computer-implemented method of claim 4, further comprising:

7. 10. The computer-implemented method of claim 1, further comprising estimating, by the system, at least one defined criterion for a quantum computer to be used to calculate an expectation of a stochastic process associated with the target probability distribution.

8. calculating, by the system, one or more errors associated with at least one of applying the quantum operations to the variationally prepared quantum state to generate a quantum state, estimating the at least one defined criterion, or calculating the expected value of the stochastic process associated with the target probability distribution.

8. The computer-implemented method of claim 7, further comprising:

9. 8. The computer-implemented method of claim 7, wherein the at least one defined criterion is selected from the group consisting of an attribute, condition, property, parameter, or configuration of the quantum computer that enables the quantum computer to achieve a defined quantum supremacy when calculating the expected value of the stochastic process associated with the target probability distribution, wherein the probability distribution comprises a standard normal probability distribution and the target probability distribution comprises a normal probability distribution.

10. 10. The computer-implemented method of claim 1, wherein the quantum operation is a fault-tolerant operation.

11. A system arranged to perform a method according to any one of the computer-implemented methods of claims 1 to 10.

12. A computer-readable storage medium comprising instructions which, when executed by a computer, cause the computer to perform a method according to any one of claims 1 to 10.

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